Tables deduced from The mortality experience of life assurance companies

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° . asa Ca i ee

INSTITUTE OF ACTUARIES LIFE TABLES.

TABLES

DEDUCED FROM THE

MORTALITY EXPERIENCE

oF

LIFE ASSURANCE COMPANIES,

48 COLLECTED AND ARRANGED BY THE

INSTITUTE OF ACTUARIES

OF GREAT BRITAIN AND IRELAND:

WITH 4X
INTRODUCTION EXPLANATORY OF THE CONSTRUCTION AND
APPLICATION OF THE TABLES,

AND AN

' APPENDIX CONTAINING A COMPLETE SYSTEM OF NOTATION

FOR LIFE CONTINGENCIES.

Published by the Authority, and under the Superintendence, of the Institute.

a

LONDON:
CHARLES & EDWIN LAYTON.

1872.

) YO 4-0. oO. Ft. 2 —U

TPac/ase 3-3- as WNR

PREFACE.

In the Preface to “The Mortality Experience of Life
Assurance Companies, collected by the Institute of Actuaries,”
published in May, 1869, it was stated, ‘‘ The preparation of
monetary and other Tables for official purposes is a work
involving so much time, labour, and expense, that the Council
of the Institute of Actuaries have decided not further to delay
the publication of the original facts, on which they will have
to be computed.”

The liberality of several Life Assurance Companies
promptly relieved the Council from all anxiety in regard to
the expense attending the completion of so extensive a work.
Their further contributions, in answer to a circular issued in
June, 1869, amounted to £605, making the total Fund
collected £1142. 5s. A List of the contributing Offices is
appended to this Preface.

As a preliminary to the determination of the classes
of monetary Tables to be computed, it was considered
important—

1. That the Tables of Mortality deduced directly from the
Observations should be graduated in such a manner
as to preserve, to the utmost possible degree consistent
with their practical use, all the original features of
the Observations.

V\Eetee ep

iV

2. That further investigations should be made into the
question of the effect of selection, with the view of
deciding whether distinct monetary Tables should be
given, applicable to assured lives from which that
effect has in a great measure passed away.

The graduation of the three Tables, H™, HF, and H™F,
was kindly undertaken by Mr. Woolhouse, whose great skill
and experience suggested to him a new method, combining
in a probably unprecedented degree adaptation for all prac-
tical purposes with close adherence to the original facts.
As this method cannot fail to be of great interest, the
Tables, as adjusted by Mr. Woolhouse, together with his
explanation of the process of adjustment, are printed at the
end of the Introduction.

In order to examine more closely into the question of
the effect of selection, a Table of Mortality was deduced
from the Observations H™, excluding the years of Assurance
© to 4; this was similarly graduated by Mr. Woolhouse. It
is designated by the symbol H™(5). The particular period
adopted as most suitable generally for exclusion was deter-
mined by the marked difference, in the classified original
data, between the first five years of assurance and the
subsequent periods.

It was then decided, after a careful consideration of the
whole subject, that Commutation Tables for Single Lives, and
Values of Annuities to 4 places of decimals, at 3, 34, 4, 44, 5,
and 6 per-cent, should be deduced from the graduated H™
and HF Tables. It was also decided that similar Tables,
at 3, 34, and 4 per-cent, should be deduced from the H™(5)
Table. These were considered sufficient to afford to Life
Assurance Companies a basis for approximate valuations of
their position and prospects independent of the influence of
recent selection. The Table H™F has not been made a basis
of calculation in the present volume.

" N

Vv

The Tables for Two Lives comprise an entire set of Joint
Life Annuities for every combination of ages from (10, I0)
upwards, at 3, 34, and 4 per-cent, as being the rates generally
sufficient for official calculations. The values of Annuities on
the last survivor of two lives for the same combinations of
age are also given at the same rates of interest, and by a
simple arrangement fill up, with a new and valuable set of
Tables, what would otherwise have been a great waste of
space in the Joint Life Columns.

The laborious and responsible task of superintending the
computation and printing of all these Tables was undertaken
in the most disinterested manner by Mr. Peter Gray, whose
name will be accepted as a guarantee of the accuracy and
consequent value of the completed results.

To the Tables themselves Mr. Gray has written an
Introduction, fully explaining the methods of formation and
various modes of their application. This renders it unneces-
sary for the Council to do more than draw attention to his
clear and able exposition: but they desire, at the same time, to
record their sense of the great obligations which all interested
in the present work are under to Mr. Gray for his invaluable
assistance in its production.

Other Tables are required to complete the Canon, such as
Single and Annual Premiums for Survivorships and Endow-
ments and the values of Temporary and Deferred Annuities,
but it appears to the Council that it would not be advisable
to delay any longer the publication of the present volume,
which, with these explanations, they now submit to the
Profession.

The Council have thought it desirable to append to this
volume a comprehensive Scheme of Notation, settled by a
Special Committee of Actuaries, after long and careful con-
sideration of the subject, aided by valuable suggestions and

vi

co-operation from various quarters. The Council believe
that this notation will be found extremely well adapted to
express all the benefits that occur in practice, and they
strongly recommend it for general adoption, in the expectation
that its use will contribute materially to advance the study
of life contingencies by promoting exactness of thought and
expression.

LIST OF CONTRIBUTING OFFICES.

British Equitable Assurance Company.
Briton Medical and General Life Association.
Church of England Life Assurance Institution.
Clergy Mutual Assurance Society.
Clerical, Medical, and General Life Assurance Society.
Commercial Union Assurance Company.
Crown Life Assurance Company.
Eagle Insurance Company.
English and Scottish Law Life Assurance Association.
Equity and Law Life Assurance Society.
Friends’ Provident Institution. |
General Reversionary and Investment Company.
Guardian Fire and Life Assurance Company.
Hand-in-Hand Fire and Life Insurance Society.
Imperial Life Insurance Company.
Law Union Fire and Life Insurance Company.
Legal and General Life Assurance Society,
Liverpool, London, and Globe Insurance Company.
London and Provincial Law Assurance Society.
London Assurance Corporation.
Metropolitan Life Assurance Society.
National Life Assurance Society.
National Union Assurance Company.
North British and Mercantile Insurance Company.
Northern Assurance Company.
Pelican Life Assurance Company.
Prudential Assurance Company.
Queen Insurance Company.
Royal Farmers’ Insurance Company.

. Scottish Imperial Insurance Company.
Sovereign Life Assurance Company.
Sun Life Assurance Society.
Universal Life Assurance Society.
West of England Fire and Life Insurance Company.
Whittington Life Assurance Company.

——

Vil

CONTENTS.

Page

PREFACE iii

InrRopucrion . ee . 1x

EXxrLaNnaTiow OF THE ADJUSTMENT OF THE TABLES . . » Lexxvil

TABLES.
L.—One Life.
*.* Under each of the three Mortality Tables, HM, HF,
and H{M(5) are given :—

First, a Table of Elementary Values, with Results
deduced from it involving the rate of mortality
only ; and,

Secondly, the following Tables involving the rates of both
mortality and interest, for each rate of interest em-
ployed, namely :—

1. Commutation Table ;

2. Logarithms of the same ;

3. Results deduced from the Commutation Table.

The rates of interest will be apparent from the following |

, enumeration :—

H*®;— 1
ELEMENTARY TABLE, AND ResuLts 2
THREE Per-Cenrt. 7
Turee anv a Hair » 15
Four ” 23
Four anp A Har ” ‘ 31
Five » 39
SIx

Vili

HF:— . ; ; ; . . ae
ELEMENTARY TABLE, AND Resutts. ; . ; . 56
THREE Per-Cent . . ; , . 61
‘THREE AND A Har » . . . . . 69
Four ” . . . . - 97
Four anp a Har 9 . . ; . . 85
Five - ; ; ; ; - 93
Six - . . . . . IO]
HM(5):— . . ; ; ; , , oo, . 109
ELEMENTARY TABLE, AND RESULTS. . . . . ITO
THREE Per-Cent . , ; , . 115
THREE AND A Har » . ; . . . 123
Four » ; . ; . . 131

IT.—Two Lives.

*,* Of Two-Life Tables the values given, deduced from
H™ only, are those of Annuities on Joint Lives and on
Last Survivors. As these, for the same rate of interest,
occupy between them a single quadrangular space, a
reference for each rate of interest is sufficient.

H™ :—Turze Per-Cent ras 21°

Turee AND A Hatr » . . . . . I71

Four ” . . . . . 203
Auxiuiary TABLES FOR THE FoRMATION OF SURVIVORSHIP As-

SURANCES . . . . . . . - 235

ConsTANTS . . . . . ; . . . . 238

System or Notation ; . . . . . ; . 239

id

INTRODUCTION,

THREE distinct Mortality Tables are, in the present work,
made bases cf computation. They are designated respectively
by the suggestive symbols, H™, H¥, and H™(), The first
two, H™ and HF, have been deduced, by a highly scientific
process of graduation, devised and applied by Mr. Woolhouse,®
from the two, similarly designated, on pages 273 to 276 of
the Mortality Experience of Life Assurance Companies, collected
by the Institute of Actuaries, published in 1869. The original
tables commence at age 0, with a radix of 10,000. The new
tables, on the other hand, commence with a radix of 100,000
at age Io, the numbers observed upon, between ages 0 and I0,
being considered too small to afford trustworthy results; and
therefore the numbers-living, in corresponding tables, do not
admit of being directly compared. It will be shown hereafter
how closely the results of the graduated tables are assimilated
to those of the tables from which they have been respectively
deduced.

As regards H™M(5), the third table which is here made
a basis of computation, there is no table in the Experience
volume to which it holds a relation corresponding to that held
by the two H™ and HF, to the tables similarly designated in
the volume referred to. It has been formed from the same data
as H™, when modified by the exclusion from them of the
experience of the first five years.of assurance. Comparison
of the results of this table with the corresponding results of
H{™ serve to show the effect of recency of selection.

The arrangement that has been adopted, in regard to the
Single-Life Tables, has been to group together, under each
fundamental table, all the tables deduced from it. It is

* Much attention has been given by Mr. Woolhouse to the subject of the graduation
of tables, A full description of his method, as applied to the Table HM, follows
this Introduction. In the application of this method every fact in the table operated upon

has its due weight accorded to it; and the effect is, that in the resulting table, while
asperities are softened down, every well-pronounced characteristic in the original is faithfully

reproduced.
6

x

believed that this arrangement will be found to be more con-
ducive to facility in the use of the tables than any other that
could be suggested.

The Two-Life Tables, which have been deduced from
H™, follow the Single-Life Tables.

I. OF THE SINGLE-LIFE TABLES.

The Single-Life Tables, deduced from the several funda-
mental tables, may be classified as follows :—

1. An Elementary Table. This contains the Mortality
Table and such deductions from it as are requisite for
the construction of the succeeding tables, and of any
others that may be proposed.

2. Results involving the rate of mortality only. These
are ~,, Jz and éy.

3. A Commutation Table.

4. Logarithms and Cologarithms of the principal columns
of the Commutation Table.

5. Results deduced from the Commutation Table. These
are a,, A, and g,.

The monetary tables, comprising 3, 4 and 5 of the above
enumeration, are given at various rates of interest. For each
of the tables H™ and HF they are given at the following six
rates, viz., 3, 34, 4, 44, 5 and 6 per cent.; and for H™(5) they
are given at the three rates 3, 34 and 4 per cent.

The tables as above classified will be now more particularly
described ; and attention will be directed to any specialties in
their form, or in the methods employed in their construction,
that may seem to deserve or to demand notice.

The illustrative references will be solely or chiefly to the
table H™ and the deductions from it, with interest, where
this element enters, at 3 per cent.

1. The Elementary Table.

The functions here tabulated for each age are /, and @,, with
their logarithms and cologarithms; and also the logarithms
and cologarithms of #,, the logarithmic functions being in all
cases accompanied by their differences, It has not been

Xi

usual, for what reason is not apparent, to tabulate log d@,; nor
have the differences been heretofore tabulated. These find
their uses, as will hereafter appear, in the construction of
tables.

The bases of the logarithmic portion of the Elementary
Table are log/, and logd,. The manner in which the
remaining columns are deduced from the columns containing
these functions is sufficiently obvious; and it is therefore
necessary to explain only the methods employed for their
verification. :

Since, #, being any function of z,

Uzpt Ate t+ Atri t . . . tHAUr a1 = Ugia,

it appears that if in any column of differences we insert at
intervals the proper values of w#,, they will in each case be
the sum of all the terms which precede them. Thus, referring
to p. 2, if in the blank spaces in the column headed logs,
(which contains the differences of log /,) we insert log dp,
log ¢,5, log fy, &c., the column will be verified by continuous
addition. But the requisite additions can very well be made
without the actual transference of the terms from the adjoin-
ing column. All the columns of differences were checked in
this way.

Again, since the sum of a logarithm and its complement
is O, so the sum of any number of logarithms and their com-
plements is alsoo. The complementary columns were verified
therefore by adding together, as they stand, the corresponding
groups of five in the two columns.

2. Results involving the Rate of Mortalsty only.

These results, as respects H™, are on p.6. They consist
of ~,, g- and ¢,, for each age. The first two are complements
of each other to unity; and they might have been formed by
taking out the numbers corresponding to log f, in the
Elementary Table, and subtracting the results from unity.
But it was preferred to employ for this purpose a method
which brings into use certain of the tabulated functions ; and
as it is typical of other operations to follow, it will be

described at some length. 7

Xil

Since 9.= a,
and consequently, log g,= log a,—log /,
| =log d,+colog Z, ;
therefore A log 7,== A log d, + A colog /,
=A log d,+colog f,,
(since A colog /,=colog 7,).
Now log 7.4,=log 9,+ A log ¢,.

Hence, by substitution,

log Jz41=log 7. + Alog d, + colog pz.
That is, we shall pass from log ¢, to log ¢,,, by adding to the
former the differences of its components ; and the logarithms
of the series g, will consequently be formed by continuous
addition.

The formation of the first few terms of the series is here
exhibited.

Lo Ps

on (l=gs)
Logdyy 2°690196
*Col. 449 5-000000

10 3°690196 ‘004900 = ‘995100
908595
2133

rr 600924 "003990 "996010
918405
1736

12 ‘321065 003319 "996681
942196
1444

13 °464705 "002915 997085
975177
| 1268

14 °441150 "002762 "997238
15680
1201

15 °458031 7002871 "997129

d @ * #
Verification.
Logd,; 2°450249
Col. bs 5-007782 .
1s 63458031
* Col. is adopted in the illustrative examples as an abbreviation for Co/og, in order to
range better with Log.

Xill

Log dj) is set down on the first line, and on the fourth,
seventh, and every third succeeding line the differences of
log d, in order, commencing with Alogd,,; colog 4 is. set
down on the second line, and the differences of colog/,, in
other words the successive terms of colog~,, commencing
with colog 1), on the fifth, eighth, &c., that is, on every third
line as before. The third, sixth, ninth, &c., lines thus left
vacant, are intended to receive the results of the final additions,
which results are the logarithms of the values sought. Before
proceeding to the final additions the terms set down should
be added continuously in groups; and the successive sums,
which should be previously formed from the Elementary
Table, and inserted in their places, will serve, as they are
reached in order, to verify also the final additions. —

In most, or all, of the logarithmic series with which we
have to do, the increase (or decrease) in the successive terms
is so gradual that it is not necessary, in computation, to write
the index of more than the first term. We have sufficient
intimation of a change in the index by that which takes place
at the same time in the mantissa of the logarithm. It is
thus unnecessary to set down the indexes of the differences,
and they are not inserted in the tables.

It is frequently desirable to form a series in reverse instead
of direct order; and the Elementary Table furnishes the
means of doing so. Thus:—

Since log g.4:=log g,+ A log d,+ colog Z,,
therefore, logg,=logg,.,—A log d,—colog p,
=log 7211+ A colog d, + log p,,

(Acolog ad, being the negative difference of logd,, marked
—A in the Elementary Table); and the series may be formed
by continuous addition, as before.

The following is the construction of the last six terms ;
and the results of this and the preceding formation may be
compared with the complete table on p. 6.

XiV

Log 7s Gz (19,)
Log dpy 0-954243
Col. yy 1-045757

97. 9:000000 __t-e00000 "000000
647817
264047

96 =. 911864 "816327 "183673
332438
559862

95 °804164 "637036 "362964
208517
692583

94 705264 507300 ‘492700
147020
766578

93 ‘618862 "415778 *584222
114799
812035

g2 545696 "351314 "648686
@ @ # #
Log dy, 2'404834
Col. fog 3140862
92 1545696
It is hardly necessary to remark that, the column g¢,
shewing the annual decrement on a unit of life at each age,
it shews also, by the proper disposition of the decimal point,
the annual mortality per hundred, per thousand, &c. Thus,
we see that at age 52 the mortality is 1? per cent., or 174
per thousand, and so on.
The next result is ¢,, the Average Duration of Life at
each age. The column has been formed by the aid of the
well-known formula,

€g=P,(1 + Cri);
in which e, denotes the Curtate Duration at age x, and from
which the Complete Duration, ¢,, is deduced by addition of *5
to each term.

The formation of the first few terms of the series (the last
few as the table is arranged) is here subjoined. The initial
term, 7 being 0, is log ég=logfo,. The operation is con-
ducted by means of the table Log (1+) in Gray’s Zables and
Formula, and verification is obtained from point to point by
the use of the formula,
frei losat 2s os

(= +.

£

xV

Log ez es Cr

97 “0000 "5000

96 1:264047 1837 ‘6837
559862
073225
__f

95 633094 "4296 "9296
692583
155195
28

94 847806 "70.44 1°2044
766578
231564
___#

93 998144 9957 _1°4957
812035
300081
22

92 112138 1°2946 1°7946

# # # #
bogtlogt .. 47 _ 936
log 723

Log936 =. 2971276
Col.723 3140862
92 0-112138

Comparison of the series thus formed with the corres-
ponding series formed from the same table, H™, previous to
graduation, Experience, pp. 281, 282, gives, for ages 10 to Qo,
the following result:—In 20 cases the corresponding terms of
the two series are identical ; in 26 they differ by +or; in 11
they differ by +02; in 12 by +'03; in 2 by +°04; in 1 by
—°05; in 3 by +06; in 3 by +'08; in 1 by +°10; in 1 by
—'I5; and in 1 by —'18.

Or it may be put thus:—The sum of the positive devia-
tions is ‘81, and that of the negative deviations is 1°01; and
we have

XVi

20 cases of coincidence ;

34 with an average deviation of + ‘024 ;
and 27 » ” » —'037.
From which it appears that the graduated table may be
unhesitatingly accepted as a faithful representation, but
deprived of its angularities, of that from which it is deduced.

A similar comparison to the foregoing, instituted between
the Average Durations deduced from the original and the
’ graduated HF tables, for the same ages, 10 to 90, gives a
somewhat less favourable result. The former is to be found
in the Experience volume, on pp. 281, 282, and the latter on
p. 60 of the present work.

Here we have :—3 cases in which the corresponding terms
are identical; 15 in which they differ by +'o1; 14 in which
they differ by +:02; 14 by +°03; 5 by 4°04; 5 by +°05;
3 by +°06; 2 by —‘07; 1 by +08; 3 by —‘09; 2 by —10;
3 by H°11; 3 by 4°12; 1 by —*13; 1 by +°14; 1 by +°16;
1 by +°18; 1 by —*19; I by +'24; 1 by —°27; and 1 by
+°43-

The totals are, of positive deviation 2°48, and of negative
2°12; and we have finally :-—

’ 3 cases of coincidence ;

38 with an average deviation of +065 ;

40 ” ” ” —'053.
The deviations are here, as already remarked, confined within
less narrow limits than in the case of the H™ table. The
cause of this is, that the original HF table contains more
asperities than the original H™; and this again arises, no
doubt, from the comparative smallness of the number of lives
entered in the observations on which it is founded. And it is
to be borne in mind that the deviations (which legitimately
arise in the process of graduation) are in the direction of
softening down the asperities by which, like other results of
original observations, the present table is characterized. The
graduated table may therefore be confidently accepted as an
exponent of the law of mortality likely to prevail amongst a
class similar to that subjected to observation.

xvii

The tables H™ and H™(5) are formed, as has been already
stated, from the same series of observations. In the case of
the former table the lives are brought under observation at the
date of assurance, that is, at the period of their selection, as
healthy lives ; in the case of the latter table they are not brought
under observation till they have been assured five* years.
Hence, if recency of selection has any effect in modifying
the rate of mortality in a community, this ought to become
apparent on comparison of the results of the two tables formed
as above described. And accordingly we find, on referring to the
Average Durations, pp. 6 and 114, that those on the latter page,
derived from the table H™(5), are throughout less than those
on the former, derived from the table H™. The difference,
which at the outset is 1°83 years, remains nearly stationary for
12 terms, after which it gradually decreases ; but it does not
entirely disappear till the close of the series. The conclusion
then is, that recency of selection exercises a by no means
unimportant influence on the rate of mortality; and the
measure of its effect in the present case is the series of
differences between corresponding terms in the two sets of
Average Durations now under consideration.

3. The Commutation Table.

We now come to the tables involving the rate of interest
as well as the rate of mortality. The principal and funda-
mental of these is the Commutation Table. It is necessary to
say, that in the Commutation Tables here given, Davies's
relation of the several columns to each other has been adopted ;
so that we always have

and soon. And hence the formule for their application are
those of Professor De Morgan’s paper in the Companton to the
Almanac for 1840, reprinted in the Fournal of the Institute,
vol. xii. p. 328. This form has been adopted after fully

® More exactly four and a half years, on the average.

XVili

considering and duly weighing the arguments that have of
late years been adduced for a departure from it, and which
need not here be recapitulated.

The methods employed in the construction of the funda-
mental columns will now be described. And first as to
Column D.

We have log D,=log /, + log v*,
and log D,,,=log /,,, + log v**' ;
therefore 4 log D,=log f, + log v
=log up,.

Hence, since log D,,,=log D,+ Alog D,=log D, + log uf,, it
appears that the column log D, may be formed in direct order
by the continuous addition to log D,) of the terms of the
series log vp,. But it is more convenient, by facilitating the
formation of Column N, to form log D, in reverse order.
From the foregoing we have

log D,=log D,,, —log uf,
=log D,4, + colog up,.
Hence, commencing with log D»;, the column is formed by
the continuous addition of colog v+colog Z,.
A specimen of the operation follows :-—

Log v* Log Ds D, and N,
100 62°716278 Loge 2-754789
99 °729115 yy Ig7 §=0°954243
98 741952 ‘709032 .
97 “754789 s7 012837 su7 9
96 767626 735953
33 a aesol 96 “457822 2°8696
012838 paRia
93 ‘806138 3°3813 95
92 818975 440138
91 8318134 95 9107984 81433
90 844650 012837 ak
Bo 857487 307417 "524094
88 = 870324 94 ‘231052 17°0236
87 = 883161 012837 38° <aBa
85 Spee 93 ‘477311 30°OI31
. 012837 TRcGla
187965 5850139
92 678113

eee ee eee 7

xix

58°5613
92 678113 47°6555
012838 Toe a1 tA
162878 106°2168 ro
g1 °853829 + 71°4215
012837 yo>-aRo
149337 *77'0383— 90
go 009003 102°095
012837 :
126794 77973389
8g -°148634 140°810
012837 ,
118786 #7054388
88 ‘280257 190°659
012837 Sr i-a0a »
114903 611°203 87
8; -407297 255°445
012838 6
107713, 84788
86 ‘527848+ 337°169
012837 3202316
1oagi1 70 BHO BS
85 °642996 439°538
1643°354 84
1d # ® a
Log 8 2-908836
‘a5 3°734160
- «Bg 2642996

The first line is occupied by log v”, and the fourth, seventh,
tenth, &c., by cologv; the second line is occupied by log 4,
and the fifth, eighth, eleventh, &c., by the successive terms of
colog g, in reverse order. Before proceeding to the additions
it is necessary to introduce certain corrections. The true
value of colog uv is ‘012837,2247 ...., and the neglect of
the figures beyond the sixth would give rise to constantly
increasing error, which, at the close in log Dj, would
amount to no less than 20. To counteract this a periodical
correction is made, and the points where it is to be in-
troduced are easily determined by reference to the series
log v*, a portion of which, in reverse order to correspond with

the order in which the table is being formed, is given in
c2

xX

the margin.* Commencing with logv® (which is the term
that enters into log D,,) and attending to the last figures only,
we see that the differences are 7, 8, 7, 7, 7, 8, 7, 7,7, 7,8, ...-
Let the terms on which the 8’s fall be marked with +, meaning
that those terms have been increased by unity. The marked
terms here are 95, 91, 86, &c., and the terms in log D, corres-
ponding to those ages will thereupon have to be similarly
increased, since log D,=log/.+logv*. The + marks are
therefore placed against the proper ages in the working
column, and it is then seen that what is requisite is to alter
the value of cology next preceding each marked line from
012837 to ‘012838. This correction being madef the addition
is proceeded with, and the successive sums are the terms of
the series log D,, true to the nearest unit in the last place.

It is obvious that the frequent writing of cology may
easily be avoided. It should be written at the bottom of a
card ; and this being held over the sum last formed, the loga-
rithm can be included in the next summation. This method
will be exemplified in the formation of log C,.

The numbers corresponding to log D, were taken out by
one computer and carefully examined by another. Further
verification was obtained when Columns N and M were formed,
as will be shewn hereafter.

The formation of a portion of Column N is shown; but
it is unnecessary to exemplify that of Column S.

The method employed for the formation of Column C
will now be described.

We have

log C,=log v,.1+ log d,.

Hence, A log C,=log v+ A log d,.

® A six-figure table of log v*, which shall be true in the last place, cannot be formed
from a seven-figure table, since we are left in doubt as to whether or not a terminal ¢ in
the seventh place indicates an increase in the sixth. And in especial it cannot be formed
from Mr. David Jones's values of this function, these being very incorrect in the last place,
particularly the 4 per-cent. column. An excellent set of tables of log ©, extending to ten
places, and therefore suited for every purpose, is given in Mr. William Thomas Thomson's
Actuarial Tables, to which work, Mr. Thomson informs his readers, it was contributed
by Mr. Filipowski.

+ The corrected figures are printed in heavier type.

Xxi
Consequently, for the formation of log C, in direct order,
we have
log C,,,=log C,+ A log C,
=log C,+ log v+ A log a, ;
and for the formation in reverse order, which is the more
convenient, we have .
log Cp=log C,,,+ colog v + A colog d, ;
in which Acologd, is the function occupying the column
marked —A in the Elementary Table.
The following is an example of the formation :—

Log C, C, and M,
Log v 3741952
» Gg, 0°954248
9 1:696195 .
F GATBIT 490891
96 °356849 .
332438 ~ 243
——__ 27711 96
208518 78076 gs
94 °'923479 + 8°3845
147020 16°1931 94
93 083336 I2°II 54
114799 383075 8
92 :210972 16°3.544
112362 44°5619 2
9: °836171 21°6856
__93465 66'°2475 g!
90 06 442.473 + 27°6995
889% 93°9470 go
89 539255 34°O143
94270 128'5613 89
88 646362 44°2957
__99304 172'8570 88
87 °758508 57°3460
85411 Ggo'a030 87
86 856751 71°9036
82558 302°1066 86
85 9521414 89°5656
391°6722 85
® d # &
Log ds 3-056142
vw 2-895999

85 1952141

XXil

The formation here is effected in the manner just referred
to. The only addends written down are the terms of the
series Acologd, (in reverse order), and colog v is included in
each summation (after the first) by means of a card.

In consequence of the constant addend, colog v, not being
set down, the corrections rendered necessary by its curtailment
to six places must in this formation be applied to the terms
of Acologa,. Their places are determined at once by
reference to the formation of log D,. In the present formation
the marked terms are those corresponding to the ages one
year younger than those to which they correspond in the last-
mentioned formation. The powers of wv which there enter
into’ D,, enter here into C,.,. Hence, applying the marks,
and increasing the addend immediately preceding each by a
unit in the last place, continuous addition gives the terms of
log C, true to the nearest figure.

The numbers being taken out and examined, the formation
of Column M is proceeded with, and a specimen of the opera-
tion is exhibited.

We can now procure verification of all the preceding
work,

Since M,=vN,,—N:,
we have My=vNg—Nes;
and the formula is applied as follows :-—

103)1043354( 1595490 oNg.

13 1203816 85°

983 301674

565 391074 Mg;
5°4
92

This process is easier than the logarithmic one, when, as
here, division by 1 +4 is used instead of multiplication by v. It
also, usually, gives us one or two figures more in the result
than can be obtained by the use of logarithms.

A similar relation holds between the Columns S and R.
Thus,

. R,= VS2-1— Sx ,
And by the application of these formule the whole of the
Commutation Tables in the present volume have been verified,
from point to point.

Xxili

4. Logarithms and Cologarithms of the Principal Columns of
the Commutation Table.

This table, which contains also the differences of the loga-
rithms enumerated, occupies the four pages next following
each.of the Commutation Tables. The foundations of the table
are the columns containing log D,, log N, and log M,; and the
manner in which the remaining columns are derived from these
is sufficiently obvious to preclude the necessity for description.
The columns were severally verified in the same way as the
corresponding columns of the Elementary Table.

5. Results derived from the Commutation Table.

The results here tabulated are @,, A, and a, the values of
annuities, and the single and annual premiums for assurance
of a unit.

They were formed as follows :—

I. ()

.*. log a4,=log N,+ colog D,.
Hence, A log a,=A log N,+ Acolog D,,
=A log N,+colog vf, ;
and log @,,,=log a,+ A log a,
=loga,+ A log N,+colog uf,.
2. Similarly :—

and log A,=log M,+colog D,.
A log A,=A log M,+colog z,.
log Ay, =log Ag+ A log A;
=log A, + A log M,+colog v,.
3. And finally :-—

_ M,
ve New’
and log #,=log M,+colog Nz).

A log #,= A log M,+ Acoloz N,_).
log #,,,=log #w,+ A log w,
=log#,+ A log M,+ Acolog N,_;.

xxiv

The following are examples of the three operations :—

Log az

6°254516
5-128372

10 | 1-382888
982268
14971

rr] °380127

982136

14573

12| ‘376836

981984

14281

13} ‘373101

981818

14105

14| ‘369024

981641

14039

15] °364704

981454

14086

16] °360244
e ®

6°145817

5°214427

16 | 1°360244

az

Log A,

4298988
§-128372

24°1484 | 1:427360

23°9953

23°8142

23°6103

23°3897

23°31 581

23°9315
®

992208
14971

434539
993771
14578

"442883
994922
14281

452086

995637
14105

461828
995961
14039

471828
995896

14086
481810

4267383
5-214427

1-481810

A;

"267523

"271981

"277257

283195

F

"289620
296366

"303256
#

Log w,

4-298988
7727862
2-026850

992208
17622

036680

993771
17732

048183
994922
17864

060969
995637
18016

074622
995961
18182

088765

995896
18359

"103020

4°267383
7-836637
2°103020

Wr

"010638
"010881
“OLII73
"OL1507
"011875
"012268

"013677
#

The initial values are log Ny» + colog Dj, log Mj) + colog Dio,
and log M,y)+colog No, the addends being in accordance with
the preceding formule. Neither colog N, nor its difference is

in the table.

Nio
Dio

Ny

A

They are easily formed, thus:—

1796867

74409
1871276 log 6'272138
col. 7°727862
017623

The verifications at the bottom are log N¢+colog Dg,

log Migt+ colog Die, and log M,. + colog Nis-

XXV

6. On the Extent of the Tables.

The limits prescribed at the outset for the extent of the
tabulated values, as sufficient for practical purposes, were as
follows :—For the logarithms, six places, throughout ; for D,
and M, of the Commutation Tables, also six places ; for the
remaining columns, N,, S, and R,, seven places; for the
assurances, six places; and for the annuities, four decimal
places. The Elementary Tables were formed by the aid of
six-figure logarithms ; but for the Commutation Tables and
the deductions from them, seven-figure logarithms were used,
the whole, logarithms and numbers, being subsequently cut
down to the prescribed limits. It may fairly be presumed
therefore that, the tables thus formed are true in their last
figures.

The object in view, in the employment of seven figures, at
the cost of considerable enhancement of labour, was the
attainment of a greater degree of exactitude than was to be
had by the use of six figures. It is well enough known that
a result formed by means of logarithms may generally be
depended on, subject to an error of a unit or so, to the same
number of places as are contained in the logarithms used in
its formation ; but it is not so well known what degree of
exactitude may be looked for in the results of such a series of
mixed operations as is required in the formation, for example,
of a commutation table.

With a view to throw a little light upon this point, six-
figure logarithms only have been employed in the portions of
the commutation table given as examples ; and comparison
of these portions with those corresponding to them in the
printed tables, reveals the existence of a few discrepancies of
a unit in the last place. But, to obtain a wider field for com-
parison, Column D, commenced in the example, p. xviii, has
been completed on the same basis, Column N has been formed
from it, and from those columns, still using the six-figure
tables, a table of annuities has been formed.

The following small table shows the result of a careful

comparison of the columns formed as just described with the
a

XXvi

corresponding columns in the tables, these latter being assumed
to be correct :—

N. a, | Totals,

48 50 77 235
39 33 10 IIo
T 3 e¢@ 4
@s6 I @e I

This means that in log D, there are 60 values correct, and
28 with an error of +1; in D,, 48 correct, 39 with an error
of +1, and one with an error of +2, and so on; the state of
the annuity column being, 77 correct, and 10 with an error of
+1; that is, in the fourth decimal place, three only being
usually tabulated.

These deviations from strict accuracy will not be considered
very great. In order to determine the cost by which the
greater exactitude of the tables as printed has been purchased,
an experiment was instituted. Three gentlemen, well prac.
tised in the use of both the seven and the six-figure tables,
kindly lent their aid. Each of them made a certain numbef
of direct entries, first in the seven and next in the six-figure
tables. A like number of inverse entries was then made,
first in the seven and next in the six-figure tables as before ;
the times occupied being carefully noted in each case. Com-
bination of the results gave the following ratios of the times
occupied in the two cases :—

Direct entries . . . . . . . I: 663
Inverse _,, rr Orie) €;

In both cases the ratio is greater than that of 3 : 2, showing
that in each there is a saving in time (to say nothing of the
saving in mental effort) of at least one-third, by the use of
the six-figure tables, the saving in the case of the inverse
operation being the greater of the two.

The practical inference fron. what precedes is that, where

XXVii

such a degree of exactitude as that which was found to have
been attained by the use of six-figure logarithms in the con-
struction of the Commutation Table is judged sufficient for
the purpose in hand, it is nothing else than an improvident
waste of time and labour to use the seven-figure tables. Had
the present investigation been made previous to the construction
of the Commutation Tables, and its result known, in all likeli-
hood the employment of seven figures here would not have
been thought of.*

7. Miscellaneous Formations.

It will be proper now to give, in the form of Problems,
some examples of more general formations, those already
given having reference to tabulated functions. The differences
of the principal logarithmic functions are given in these
tables for the first time; and some further justification of
this departure from previous usage than has yet been afforded,
will perhaps be looked for.

PROBLEM 1. To construct the series log vf,.

This series presents itself as the series of differences of
log D,. It is frequently wanted however when log D, has not
been formed, as for the construction of a table of annuities,

* There is at present no English six-figure table, properly so called, available ; which is
the reason why the late Professor De Morgan, as he informs us,f on occasions when he
suspected five figures to be insufficient, immediately had recourse to a seven-figure table,
We have abundance of tables in which the logarithms extend to six places; but as the argu-

“ments extend to no more than four, they are more troublesome to use than the seven-figure

tables, with a figure fewer in the results. The differences in them range from 434 to 43,
and for want of room there are no proportional parts, while there are two figures to be
proportioned for. A proper six-figure table, on the other hand, has, like our seven-figure
tables, five-figure arguments, and consequently gives five-figure results by inspection; the
differences range from 44 to 4, affording ample space for the exhibition of the requisite
proportional parts; while, moreover, there is never more than one figure to be proportioned
for.

It is hoped that ere long an English Table, with various improvements for facilitating
its use, will be available. In the meantime a very good German table, possessing the
above-described qualifications, and whose title follows, may be had at a very moderate
cost :—Logarithmisch-trigonometrische Tafeln, mit sechs Decimalstellen. Vom Dr. C. Bremiker,
Berlin, 1869.

+ Penny Cyclopedia, vol. xxiii, p. 499; and English Cyclopedia, Division, Arts and
- Sciences, vol. vii., Col. 1005.

XXVIII

for instance, and therefore the method of its formation will be
here shown.

Since log vp,=log uv + log 2,,
therefore, A log vp,= A log p,.
~ Hence, log vf,,,=logvp~,+Alogp~,; . . . , (1)
and, log 7,=log vp,,, + Acolog p, . . . (2)

Of the two formula deduced (1) is for the direct, and (2)
for the reversc formation. The latter is the more convenient
for two reasons: first, the series is usually wanted in reverse
order ; and, secondly, the differences which form the addends
in this order, have many fewer significant figures than those
that form the addends in the other. An example of each
formation is given :—

Log tp,
Logv ‘987163 Logv ‘987163
1 Pro ‘997867 » Pog °264047
to °985030 96 251210
397 295815
11 °985426 95 °947024
292 132721
12 °985719 94 “679746
176 73995
13 °985895 93 °753741
67 45457
14 *985962 92 799198
999952 25087
15 ‘985913 g1 *824284
° ° ° °
Logvy -987163 Logy -987163 —
» P15 "998751 » Poi 837122
15 °985914 gr °824285

This series, in whatever way it is to be used, whether in the
formation of Column D, or in the construction of a table of
annuities, requires correction in consequence of the curtail-
ment of the value of log zv, which enters every term. The
true value of logy is ‘987162,775 ...., the values used being
"987163 ; and the correction is the abatement of a unit in the
sixth place at the proper points, which may be determined, as
already shewn, by refererice to a six-figure table of log v’*.
The requisite corrections are introduced in the example.

XXi1xX

PROBLEM 2. To form a table of loaded assurance premiums,

The formation of w, has been already exemplified. The
present formation differs from that referred to in no other
respect than that here the logarithm of (1 plus its loading) is
included in the initial term.

Thus, to form #, with a loading of 174 per-cent. :—

Log 1175 0:070038
rT) Mio 4°298988
Col.Ny 7°727862

10 §62°096888 "012499
992208
17622

1r *106718 012785
993771
17732

12 °118221 "013129
994922
17864

13 *131007 "013521
* * °
Log 1°r75 0-070038
» Mis 4:279889
Col.Nj, _7°781080
13 2°131007
The premiums here formed may be compared with those
for the same ages on p. I4.
PROBLEM 3, To form a table of the values of endowments
payable at a specified age.
The age at which the endowment becomes payable being
z+n, the value of the benefit is
Desa.

’
#

or, in logarithms,
log Dz.,+colog D,. .
And the difference of this expression, in which «+ is
constant, is
A colog D,, or colog vf,,.
Two examples follow, in which «+ takes the values 21
and 60, respectively.

Values of Endowments.

Payable at 21, Payable at 60.

LogD,, 4:710940 | Log De 3-999631
Col. Dig 5°128372 | Col. Dig 5°128372
10 61-839312 690736 | ro 61:128003 = +1342
14971 ? ! 14971 4070
11 *854283 714962 rr = 142974 "13898
14573 14573 9 7
12 °868856 739360 12 °157547 "143730
14361 | ca,
13. 883137 764057 13 ‘171828 —_-148535
_ 14105 _ 14105
14 °897242 789300 14 185933 "153438
14039 14039
15 911281 815232 15 ‘199972 "158479
14086 14086
16 925367 842106 16 °214058
e ° e ! # e ®
LogD,, 4°710940 | LogDg 3-999631
Col. D,, 5°214427 | Col Dig 5214427 +1647 04
16 1:925367 16 1:214058

PROBLEM 4. To form a table of the assurances equivalent
to a present value of a unit.

The value of this assurance, when the age is 2, is,

Dz.
M,’
the logarithm of which is,
log D,+ colog M,.

And the difference of this expression is,
log up, + A colog M,.

The required table may be consequently constructed as
follows :—

XXX!

Assnrances whose value 1s a Untt.
Log Dip 4871628
Col. Myo 5°701012

10 6 0:572640 3°7380
985029
7792

11 ‘365461 3°6767
985427
6229

12 °557117 3°6068
985719
5078

13 °547914 3°5311
985895
4363

14 ‘538172 3°4528
985961
4039

15 ‘628172 3°3742
# 2
Log Dis 4:799659
Col. M)s 5°728513
15 0:528172

The use of such a table as that of which the formation is
here exemplified is to facilitate the conversion of a cash bonus
into a reversionary bonus.

Example. A policy on (15) has assigned to it a present
bonus of 42°375. Find the equivalent reversionary bonus.

42°375 X 3°3742= 142°982.

This table could be very readily formed from the Assurances
on p. 14 by means of Colonel Oakes’s Zable of Reciprocals.
The method here employed possesses the advantage of giving
also the logarithms of the values formed.

PROBLEM 5. To form a complete table of the values of
deferred and temporary annuities.

By a complete table is to be understood a table, that shall
comprise, in regard to the classes of annuities specified, all the
cases that can present themselves in the use of the mortality
table employed.

dr,
XXXili

The present age, (or, preferably, the age to which the value
to be formed has reference,) being as usual 2, denote that at
which the deferred annuity is to be entered upon, and the
corresponding temporary annuity to cease, by y. Then, the
value of the deferred annuity being

N,
D,
of which the logarithm is,
log N,-+colog D,,

we shall have to form the values of this expression for all the
combinations of « and y, in which x does not exceed y._ This
can be very commodiously done in a series of paralel collumns.
Either of the quantities « and y, as may be arranged, will vary
in the columns; and the other will vary in passing from
column to column, that is in the rows. If, as is most conve-
nient, it is chosen to commence with the younger ages, the
work will assume one or other of the following forms :—The
first, if in the columns zx be made to vary, and the second, if
y be made to vary :—

10 96 10 96
10 10 10 10

96 96 9 96

The forms are theoretically cqually eligible. The first,
however, is to be preferred, because in it the addends will
consist of fewer significant figures than in the second:
A colog D, contains generally only five significant figures,
while A log N, contains six throughout. )

The following is a specimen of the formation, in which the
values of y are at the top and those of z at the side.

XXXill

Formation of Deferred and Temporary Annusties.

Io

It

[2

24°1484 24°1484 24°1484
10 (382888 | 24°1484 | 365156 | 23°1823 | 347292 | 22-2480
14971 | 0-966: | 14971} 1:9Q004
23°9953 23°9953
KI 380127 | 23°9953 | 362263 | 23°0284
14573 | 09669
23°8142
r2 376836 | 23°8142
Formation—(continued).

13 14 15
24°1484 24°1484 24°1484
10 | 329276 | 21°3440 | 311094 | 20°4688 | 292735 | 19°6216
14971 | 2°8044] 14971} 3-6796] 14971] 4°5268
———_—— | 23°9953 23°9953 | —_— | 23°9953
11 | 344247 | 22°0926 | 326065 | 21°1868 | 307706 | 20°3098
14573 | 179027 14573 | 2°8085 | 14573) 36856
23°8142 23°8142]_ ss | 23°81 42
12 | 358820 | 22°8465 | 340638 | 21°9098 | 322279 | 21:0029
—___ | 23°6103 23°6103 23°6103
13 | 373101 | 23°6103 | 354919 | 22°6422 | 336560 | 21°7050
14105 | o9681] 14105} 1:9053
23°3897 23°3897
14 369024 | 23°3897 | 350665 | 22°4215
14039 0°9682
23°1581
15 364704 | 23°1582

The first six columns are here given complete. The initial
terms are the successive values of log N,+colog D,, when y is
made to vary; and they are formed as follows, the addends
being the terms of Alog N, :—

XXXIV

Log Nio 6°254516
Col. Dig 5°128372
10 1°382888

982268
rr - °365156

982136
12 347292

981984
13 329276

981818
14 311094

981641
15 292735
*% *%

Log Nis 6164363
Col. Dio 5-128372
15 1292736

In the columns, x being the variable, the addends are, as
already intimated, the terms of A colog D,, or colog uf,, com-
mencing in each column with colog vf,>, and continued till the .
next value of x at the side is the same as the value of y at the
top. °- The addends having been written down, it is well
before proceeding to the additions to form series for verifica-
tions of the several columns at suitable intervals. These will
be formed in the same way as the series of initial terms. The
following is an example for x=12 :—

Log Nj, 6-218920
Col. Dig 5°157916

12 1:376836
981984

13 358820
981818
14 340638
981641

rs ‘322279
#

#

Log Ni, 6164363
Col. Dp 5:157916

1s 1322279

The verification values being inserted in their places the
addition is proceeded with. A final verification is had at the

XXXV

close, the last sum in each column being the logarithm of the
whole-life annuity on the age at the top and also at the side.
The manner of forming these logarithms has been already
shown (p. xxiv).

To pass now to the columns of numbers. It will be
observed that in each column the lines next above the values
of the deferred annuities are occupied by terms of the series
of whole-life annuities (from p. 14), commencing, in each
column, with the annuity corresponding to the age opposite,
viz., 10, and ending with that corresponding to the age at the
top.

Subtraction now gives the values of the complementary
annuities ; 2. ¢, the temporary annuities which, together with
the deferred annuities that form the subtrahends, are equal
to the corresponding whole-life annuities that form the
minuends.

The two sets of results may be arranged in a table of the
following form :—

Values of Deferved.and Temporary Annutties.

20°4688 | 19°6216

21°1868 |20°3098 | 1
21°9098 |21'0029 | 2
22°6422 |21°7050 | 3
22°4215 | 4

°9682 |————| 16

The horizontal lines separate the deferred from the tempo-
rary annuities, the former occupying the portion of the table
above the lines, and the latter the portion below.*

For the deferred annuities, the present ages are at the
side, and the ages at which the annuities are respectively to

* The spaces occupied by the horizontal lines might be legitimately occupied by the
whole-life annuities. But these are already given elsewhere, and the lines more effectually
serve the the purpose of separators.

XXXVI

be entered upon are at the top. Thus, the value of an
annuity on (12), to be entered upon at 15, is 21:0029, and
SO On.

For the temporary annuities, the present ages are at the
top, and the ages at which the annuities are respectively to
cease, are at the side. Thus, the value of an annuity on (12)
to last three years, and consequently to cease at 15, is
2°8113.

These two annuities being complementary, their sum,
238142, is the value of the whole-life annuity at 12. Inspec-
tion will show how the values formed in the example arrange
themselves in the table.

The values in the two portions of the table are related as
follows :—Take 12,12, for ‘example; the values that follow
vertically from this point, are respectively complementary to
those that follow horizontally. Thus, the sums of the pairs,

22°8405 + °9677,

21°9098 + 1°9044,

21'0029 + 2°8113,
are each equal to 23°8142, which is the value of the whole-
life annuity at 12.

Deferred annuities in which the period of deferment is the
same, say 2 years, are to be found ranging diagonally down-
wards, commencing opposite #, and under ++. Thus the
series of annuities deferred three years, (#=3) are 21°3440,
21°1868, 21'0029, &c.

In like manner temporary annuities, whose duration is
the same, # years, are also found, in their own compartment
of the table, ranging diagonally downwards, commencing under
x and opposite +2. Thus, the annuities to last ‘hree years
are 2°8044, 2°8085, 2°8113, &c. -And the annuities in these
two sets are complementary, each to each; since the initial
value of z is the same in both sets, and also the value of x.

Finally, it may be mentioned that the foregoing table, if
completed, would occupy exactly the same space as one of
the Two-Life Tables, namely 30 pages.

XXXVI

It is obvious that such a table as that just described would
be of much utility for the formation of temporary premiums,
which are frequently required in the case of endowments
and other benefits.

PROBLEM 6. To construct a table of the values of policies.

The value of a policy of assurance is the excess of the
value of the sum assured over that of the premium.

The sum assured being a unit, and the premium P,, where
y is the age at which the policy was effected ; then, the present
age being +, the value of the policy is,

M,—P,N,_1
*V I += = Do
=D,—(1—v)N,_1—P,N,1

\N
=1—(1—v+ P,)—=".
| ” Dy

Putting for ae its value, 1+4,, and denoting (1—v+ P,) by

Q, this expression may be conveniently written,

I—O,(l4+as) ee ee (1)

To construct the required table then we shall have first to
form the values of Q,(1+4,) for every combination of + and
y in which # is not less than y, and then to subtract the indi-
vidual results from unity. The formation here requisite, is
entirely analogous to that in Problem 5. In both it is similar
series of the products of two factors that have to be formed.
There, however, the logarithms of the factors were given,
while here their determination is a preliminary part of the
process.

The premium P, may be either the pure premium, that is
the premium at the age y given by the table used in the valua-
tion ; or it may be the office premium, in its integrity, or any
how modified. There are thus two cases, which will be dealt
with separately.

* The usual symbol for the value of a policy is V,|,, where x is the age at which the
assurance was effected, and 2 the number of years since elapsed. This is at variance with
the convention by which x represents the present age. For this reason, and to facilitate the
investigation, the symbol is here modified as in the text,

XXXVI1i

Case I. Let P, be the pure premium at age y. Then
M
P= ®) =H

=D,—(1—v)Ny1

=r _ (1-1)

N,-1
awe! Oy
“Tha, (1~—v).
Substituting this in Q,, (1) becomes,

and the preliminary operation here will be the formation of
log (1+4,), which will give us also colog(1+a,), since y
takes no values different from those taken by z.

A specimen of this preliminary formation follows :

Log(1-++-a,) A Colog(1-+-a,) A
*LogN, 6-272138
Col. Dig 5°128372

10 1:400510 997349 ©=—-2°599490 = 002651
982378

14971

11 «397859 = 996841 602 141 003159
982268
14573

12 ‘394700 996417 "605300 003583
982136
14281

13 °391117 996089 60888 3 003911
981984 ;
14105

14° °387206 995857 612794 004143
981818
14039

15 *383063 995727 616937 004273
981641
14086

16 °378790 "621210
Ps Ps

LogNis 6°164363
Col.D,, 5:214427
16 1-378790

* See page xxiv.

XXXI1X

After the examples already given, the above does not seem
The addends are A log N,1+

to require explanation.
A colog D,.

Observing that A colog (1+.a,) contains fewer significant
figures than A log (1+4,;), y, in the construction, is made to
vary in the columns; z, therefore, will vary in the rows, and

the initial terms will be formed as follows :—

Log (1+ a4) 1-400510

Col. i, 2°599490
10)6©=—- 9000000
997349

II 997349
996841

12 994190
996417

13 990607
996089

14 "986696
995857

ts °982553
* Sd

Log (1-+-;5) 1:383063
Col. (1-449) 2°599490
15 1982553

A specimen of the formation, consisting of the first six
columns, is now given, in which the values of # are at the top

and those of y at the side.

Formation of the Values of Poltctes.

10 II
10 | 000000 | 1:00000 } 997349 | -99391
"00000 2651 | -oo609
it 000000 | 100000
. "00000

7

994190
2651

996841
3159

000000

12

98671
"01329
"99275
00725
I°00000
"900000

xl

Formation—(continued).

13 14 15

10 | 990607 | -97860 | 986696 | -96983 | 982553 | 96062
2651 | ‘oz140 2651 | *03017 2651 | °03938
tr | 993258 | -98460 | 989347 | :97577 | 985204 | “96650
3159 | ‘o1540 3159 | °02423 3159 | °03350
12 | 996417 | ‘99178 | 992506 | ‘98289 | 988363 | °97356
3583 | °00822 3583 | ‘o1711 3583 | °02644

13 | 000000 | rro0000 | 996089 | ‘99103 | 991946 | “98163 '
‘00000 3911 | *00897 3911 | °01837
14 000000 | 1'00000 | 995857 | “99051
"00000 4143 | ‘00949
15 000000 | r°00000
“00000

The logarithmic operation may be verified from point to
point as in last problem. The following is the verification of
the row in which y=12.

Log (1+4,,) 1:394700

Col. ,, 2°605300
12 0:000000
996417
13. 996417
996089
14 "992506
995857
15 ‘988363

# #

Col. (14,3) 2°605300
Log (1+4,5) 1-383063
15 1-988363
A final verification is had at the close of the work, the
last sum in each column being ‘ooo000, which is what
log (1+ a4,)+colog (1+4,) becomes when +=y.

When the numbers correspondihg to the several logarithmic
series have been taken out, and their complements to unity
formed, these last, which are the policy values required, may
be abstracted and arranged in a table in either of the follow-
ing forms :—

xli
Values of Policies—Form No. 1.

II 12 13

"00609 | °01329 | ‘02140
"00000 | *00725 | ‘01540

00000 | °00822
"00000

Values of Policies-—Form No. 2.

2 <0) 1J 12 13 14 15

Io} *00000 Xo)
Ir] *00609 | *00000 II
12] *01329 | *00725 | ‘oo000 12
13] *02140 | ‘01540 | °00822 | ‘00000 13
I4| “03017 | °02423 | ‘oI1711 | ‘00897 | ‘o0000 14
15| °03938 | °03350 | °02644 | 01837 | ‘00949 | ‘00000 | 15
* * * * * * * ®

In the first form the present ages are at the top, and the
ages at which the policies were severally effected are at the
side; in the second form the ages are interchanged, the
present ages being at the side, and the others at the top. In
both cases, the difference between the two ages to which a
specified value corresponds, is the number of years that the
policy whose value is in question has been in force.

The second form seems, on the whole, the preferable one.
But whichever is chosen, a large portion of the columns will be
left blank. The blanks might be filled by the values of some
other function, say the logarithms of the policy values; and
the two sets of functions might be rendered more distinguish-
able by being written in differently coloured ink, or, if printed,
by being set up in different type, as in the Two-Life Tables of
the present volume.

The table, if completed as suggested, would occupy thirty
pages.
ft

xlii

Case 2. Let P, be the office premium, or any modification
of it.

It was found in the last Case that Q,, or 1—v + P,, reduced
to (1+4,)"', so that the expression for the policy value
assumed the form

I+,
~ T+ay

In the present Case Q, admits of no such simplification ;
and therefore the expression here to be dealt with is

I —-Q, (I + &z);
in which Q,=1—v-+ P,, and P, has a given value. The nega-
tive portion of the expression is, however, of the same form as
the corresponding portion of the expression in the first Case,
and the mode and form of its construction will be analogous.

The preliminary formation here will be, since log (1+ ,)
is already formed, that of log Q,, which, in practical applica-
tions, will be a very easy matter, since P, for all ages will be
given. For the present purpose, however, it is necessary to
form P,. Premising that we assume for this the pure premium
loaded to the extent of ten per-cent,® the formation can be
easily conjoined with that of log Q,, as follows :-—

Log P, P,and Q, Log Q, A
Log i'1o = 0°041393 =-——_——_
» My 4:298988 |-029126|
Col. Ng 7°727862 .
10 2068243 -o11yoo. |
992208 -os0828 2°610958 002831 10
17622
rr °078073 = ‘or1969
993771 041095 6138789 003390 411
17732
12 ‘089576 -o12291
994922 = -o41417 ‘617179 = =003831 12
17864
13 ‘102362

- © Ten per-cent. will be considered but a small loading for an office premium. It
serves our present purpose, however, as will be seen presently, better than 2 heavier one.

xiii
13 °102362 = -012658
995637 041784 621010
18016
™ ‘116015 -013062
995961 :042188 °625189
18182 .
15 130158 °013495
. 995896 042621 629624
18359
16 °144413 ‘013945
043071 634185
* * * e
Log i'1o §=0°041393
» Mig 4267383
Col. Nig 7:835637
16 2-144413

004179

004435.

004561

13

14

‘The above needs little explanation. Log P,=log (1°10M,) +
colog N,_;, is first formed, the addends being A log M, and
A colog N,_;.. The numbers being taken out are each increased
by 1—v='029126, thus forming Q,. The logarithms of Q, and

their differences complete this preliminary formation.

The following shows the construction of the first six
columns of the final formation, the values of + being as before

at the top, and those of y at the side.

Formation of the Values of Poheies.

10 II
400510 397859
610958 610958
ro | 011468 | = 102676] 008817 | 102051
— ‘02676, 2831 |— ‘02051
011648 | 1°02718
iT — 02718

12
394700
610958
005658 | 1°01311
2831 |— ‘°o131I
008489 ; = 1°01973
3390 |= °01973
011879 | 1°02773

— °02773

xliv

Formation—(continued).

13 14 15

391117 387206 383063

610958 610958 610958
10 | 002075 1'00479| 998164 "99578 994021 "98633
2831 |— ‘o0479] 2831 00422) 2831 "01367
rr | 004906 | 1:01136] 000995 | 100229] 996852 "99278
3390 |— 01136] 3390 |— ‘90229} 3390 00722
12 | 008296 T°01929 004385 I‘O1OI5 000242 100056
3831 |— -o1929] 3831 |— ‘orors} 3831 |— *00056
13 | 012127 | 102832) 008216 | rorgro} 004073 | roog42
— 02832] 4179 |— ‘org10] 4179 |— ‘oog42
14 012395 | r:02895] 908252 | 101918
— 02895 4435 |— 01918
16 012687 | 1°02964
— '02964

The only point in respect of which the logarithmic part
of this formation differs from that in Case 1 is, that here the
addends are the differences of log Q,. The operation may be
verified horizontally at any point, as in Case 1; but we have
not here the final verification that we have there. The last
sum in each column is no longer 0: it is necessary, therefore,
to form a series consisting of the terminal values of the several
columns in order. This may be done as follows :—

Log [QAI i tae)
Log (1 Law) * ‘400510
» Qio 2610958
10 0:011468
997349
2831
II 011648
996841
3390
12 011879
996417
3831
13 012127
996089
4179

Ig 012395

xiv

14 012395
995857
4435

15 012687

Log (1 +15) 1 383063
>» Qay-«—«2°629624
15 0-012687

Both and y here vary: the addends, therefore, are A log
(1+4,) and AlogQ,; and the terms thus formed should be
inserted in their places at the outset.

It will be observed that, with the exception of three, the
policy values here formed are all negative; and a very slight
addition to the loading would have rendered the three excep-
tions negative also. With so small a loading as ten per-cent.
a policy effected at 10 has no positive value till 14, when it is
worth just one penny. Yet there are not wanting those who
maintain that every policy, whatever may be its duration, has,
or ought to have, a surrender value; and that, consequently,
the abandonment of a policy, under whatever circumstances,
is a source of profit to the office.

II, OF THE TWO-LIFE TABLES.

The Two-Life Tables in the present volume comprise, first
a complete set of the values of annuities on the joint duration
of two lives, at each of the rates, three, three-and-a-half and four
per-cent. ; and, secondly, a similar set of annuity values, and at
the same rates, on the last survivor of two lives. These, as
has been already intimated, have been deduced from the H™
table.

The two classes of tables will be treated in order; and, as
before, the illustrations will have reference to the rate of three
per-cent.

1. Of the Foint-Life Tables.

The tables were constructed by aid of the well-known
formula,

xvi

Agy=Upey (I+ Gory) 5
which is very simply deduced as follows :—

The annuity on (#y) consists of two portions, that having
reference to the first year, and that having reference to all the
years after the first, of the possible joint duration of
(~) and (y).

The value of the first portion is vf,, and that of the second
portion is upsy 2ei1-y41; Whence the total value is,

Azy=Uey (1+ Goprytt) 3
or, in logarithms, log @,.,=log upzy + log (1 + @e41-y41):

For the application of this formula we want the values of
log up, for the requisite combinations of # and y. These
are, if for convenience we make + the older age, the combinations _
in which each value of x is connected in succession with all the
values of y that do not exceed it. The formation of the values
of this function was, therefore, the first step ; and the following
is a type of the arrangement adopted :—

eye |e eee] oe |

The values of z are at the top, and those of +—y at the side.
The former proceed from 96 to 10, and the latter from 0 to 86.
Hence + is constant in the columns, and +—y is constant in
the rows. It is the fulfilment of this latter condition that has
been mainly kept in view in devising the arrangement. It is
the series in which 7—y is constant that are wanted in the
final construction ; and they are most conveniently abstracted
when formed in rows.

The following is a portion of the actual formation :—

zZ—¥! 96 95 94 93 92 gI
© | 515256 | 106887 | 372329 | 520319 | 611232 | 661408
295815 | 132721 | 73995 | 45457 | 25087 | 20541
1 | 811071 | 239608 | 446324 | 565776 | 636319 | 681949
132721 | 73995 | 45457 | 25087 | 20541 | 15543
2 | 943792 | 313603 | 491781 | 590863 | 656860 | 697492
_73995 | 45457 | 25087 | 20541 | 15543 | 8008
3 | 017787 | 359060 | 516868 | 611404 | 672403 | 705500
46457 | 25087 | 20541 | 15543 / 8008 | 4583
4 | 063244 | 384147 | 537409 | 626947 | 680411 | 710083
25087 | 20541 | 15543 8008 4583 6490
5 | 088331 | 404688 | 552952 | 634955 | 684994 | 716573
20541 | 16543 | 8008 | 4583 | 6490 | 5402
6 | 108872 | 420231 | 560960 | 639538 | 691484 | 721975
s o s o eo 2 rT]
e 2 # # # eo 2
80 | 249797 | 545776 | 678545 | 752473 | 797753 | 822550
163 | 48 | 999933 | 999824 | 999708 | 999603
8: | 949960 | 545824 | 678478 | 752297 | 797461 | 822153
48 | 999933 | 299824 | 999708 | 999603
82 | 250008 | 545757 | 678302 | 752005 | 797064
999933 | 999824 | 999708 | 999603
83 | 249941 | 545581 | 678010 | 751608
999824 | 999708 | 999603
84 | 249765 | 545289 | 677613
999708 | 999603
8s | 249473 | 544892
999603
86 | 249076

The construction was effected as follows.

Formation of Log psy.

xl vii

be formed is log uf,.,, which is equal to
log v+log p,+log p, ;

The function to

and, connecting the constant, logy, with logs,, it will be
written thus,
log uf, + log p,.

Now if in this expression both z and y vary, as in the
rows, we have,

xi viii

Az, log up.y=A log up, + A log f,;. - - (1.)
if y alone vary, as in the columns, we have,
A, log vpey=A log~,;.-.-+. ~~. (2)
and if # alone vary, as in ascending diagonally, we have,
A, log up,y=A log up, . » « (3-)
From (1) we get, for construction of the initial and verifica-
tion series, in the rows, proceeding from left to right,
log upg y==log upssi»41 + Acolog up, + A colog J,;
from (2), for the principal formation in the columns,
log up.-y=log vpeyi1 + 4 colog J,;
and from (3), for the construction of the series of terminal
values,

log wp ey = log wp g41-4+ A colog p,.
The following are examples of the application of the first
and third formule :—

Initial Values. Verifications.
f—y=0 2—y=6 Terminal Values.
Log tpyg 1:251209 | Logepy, 1-251209 | Logepes 1:251209
» Pos 1°264047 | 4, poo 1°857663 | » Pio 1:997867
96°96 2°515256 96°90 1°108872 96°10 ©1:249076
295816 295816 295816
295815 15543 gs10 544899
95°95 1:106887 95°89 420231 132721
132721 132721 ——_—
132721 8008 | 94°10 ah oe
94°94 ‘372329 94°88 °560960 ——
73995 73995 93°10 °751608
73995 4583 45456
93°93 °520319 93°87. °639538 g2'10 §=—. °797064
45456 45456 | 25089
45457 6490 orto °822153
92°92 = °611232 92°86 °691484 _ _—=—s
25089 25089 | Log eps, 1824286
25087 5402 Pio 1997867
91-91 661408 91°85 721975 or10 ©1-822153
Log yp, 1:824286 | Log upp, 1-824286
» Po 1°837122 | ,, pes 1°897689
g1or 1661408 92°85 1721975

xlix

As appears by the formule the differences are here used in
reverse order. Theoretically A colog vf, is the same thing as
A colog f,, since v is constant. Practically, however, in conse-
quence of the unavoidable inaccuracy in log v, these functions
will occasionally differ by a unit (sometimes two) ; and there-
fore, since Acolog vf, is not tabulated, in using A colog f, in
its stead, the last figure must be corrected, by reference to the
column colog vf,, pp. 10, 12, before proceeding to the additions.*
The corrected figures are here printed in different type.

The main formation illustrates the application of the second
of the foregoing formule, and the whole of the results just
obtained will be found in their places in the portion of it given
on p. x\vii.

It will be observed that it is the same series, colog g,, (ir
feverse order,) that forms the addends in all the columns;
and consequently, when it has been inserted in the first
column, it was an easy matter to transfer it thence to the
other columns in succession. |

It may be mentioned that after the initial and tWo of thred
verification series had been inserted, the entire formation of
this function did not occupy more than twelve hours.

A specimen of the annuity construction occupies the fol-
lowing two pages, and to it we now refer.

The arrangement is here obvious from the indications at
the tops of the columns and at the side. In the columns
x—y is constant, which is necessitated by the requirements of
the working formula, and y, the younger of the two ages, is
constant in the rows. The result of the arrangement is that
the values which are to occupy the rows in the table in its
final form are here formed also in rows.

* See page xxvili:
£

]

* Formation of az-y.
y | #-y=0
96 | 915256 +0328
106887 |
013989
8

372309 239608!

053873. 027233
9 64
94 [426211 +2668] 266845,
520319 446324
102711: 073660:
2: 7
93 | 623032 -4198/519991 350176
611232 565776”
152215. 124196
9 23:
92 | 763456 -5800) 689995 —*4898/ 579549
661408 636319:

198647 173089
21 31,0 14
gr | 860076 +7246 809439 "6448) 730678! +5379
* e ! ® e * | ® ®
The same.
y 2—y=81 2—y==82
15 | 249960, |
545824, |
071067! |
a

14 |616900, -4139) 250008: +17

545757

~

150420)
___9
13 828898 O74 mote
752297, 1678302;
223812 150391
39 12
12 976148) _ 9466 828705) °67
797461! 752005
239244 223772
24 2
TI 080729! TaaT0 975779] 9458 828358
822153) 797064
346540 289050
16 39)

086153 s-asg

|

* The termin‘! portioas of these six (double) columns is given hereafter, pp. Ixiv and Ixv.

li

Formation of a,,—(continued.)
y | #—y=3 a—y=4 2—y=5 |

eS |

"1157

129702 088331) __*1226
z—y==86

ES | SS ES NE ND | RS

inne! lane

rr |616336| +4134] 249473) +177
677613 544899
150244 070991
1} 11

lii
The initial terms are formed as follows, In the equation

Agay=OP gry (I+ Aes ryt),

if z is 96, (the next to the oldest tabular age,) @24)-54)
vanishes, and we get
Agg-y = UP o6+y 5
whence
log @og-y= log Upgg-ye

From this it appears that the initial terms of the several
columns are the terms of the series log uf,., which occupy the
first column, (that headed 96,) in the formation on page xlvii.
The initial terms being then inserted in their places, each is
followed, in its own column, by the succeeding terms of the
series, (7—y constant,) of which it forms the leading term.

The logarithmic operation now commences ; and the first
column, (x—y=0,) is referred to for illustration. As shewn
by the working formula, the chief part of this operation con-
sists in the formation of log (1+4,.,) from loga,.,, which is
known. This of course can be done by aid of the common
logarithm tables, used first inversely and next directly. But
it can be done much more easily by the use of Table I. in
Gray’s Tables and Formule, which table, giving the values of
log (1 +.) corresponding to successive values of log z, enables
us, by a single direct entry, to pass at once from the given to
the required logarithm. The opération was conducted, there-
fore, by means of this table.

The method of proceeding is so simple as ‘hardly to need
explanation. The table is entered with the initial term
2'515256, which is the logarithm of dogo, and the result
°013989+8, (8 being the pro-parts,) is set down as shewn.
Addition then gives the annuity value on the next younger com-
bination, (95°95). Here, observing that 120884 is less than
515256, we know that an increase of a unit has taken place
in the index. The next entry accordingly is made with
1°120884, and an operation similar to that just described gives

litt

log ag4.9,, and soon. Having always sufficient warning of an
increase in the index by the decrease that at the same point
takes place in the mantissa, it is found to be unnecessary to
set down the indices, whereby some writing, as well as space,
ig saved,

The operations in the remaining columns being in .all
respects similar to those in the first it is not necessary further
to refer to them. From and after column 4, (x—y=3,) the
index of the initial term is 1, and in none of them does a change
take place till a considerable number of terms has been
formed.

When the logarithmic part of the operation has been com-
pleted there remains the taking out of the numbers. It has
been usual heretofore to tabulate annuity values to three
decimal places only. It is, however, now well enough under-
stood that such annuity values “ are not sufficiently exact for
survivorship questions, the results of which depend on the
differences of nearly equal annuities”*; and therefore it was
determined that the annuity values in the present volume
should be given to four decimal places. The additional
labour involved in this determination is very small indeed,
So long as the values do not exceed 10 the numbers can be
obtained to the required extent by inspection, by the use of
the common seven-figure tables, or Bremiker’s six-figure
table. For values from 10 to 20 we can now also obtain them
by inspection by the use of Mr. Sang’s recently published
seven-figure table. In this table the numbers extend to
200,000; and therefore in the portion from 100,000 to 200,000
the argument consists of six places. It is consequently in
only the comparatively small number of cases in which the
value of the joint-life annuity exceeds 20, that interpolation
in taking out the numbers is necessary; and, as only one
figure has in those cases to be interpolated for, if Bremiker’s

* De Morgan, Companion to The Almanac, 1842, p. 7; and Sournal of the Institute,
vol, xiii, p. 136.

liv
table be used for the purpose the operation is a very easy

one.

There will thus be small excuse for the restriction of
annuity tables that may hereafter be published to three
decimal places.

The following diagram will aid in the comprehension of
the final arrangement of the results :—

JOINT~LIFE ANNUITIES

telat eC tea eens -)

a ESaaununmwvenwweeeeve es
a ree

40 396

There are two sets of results to be provided for, the joint-
life annuities and the last-survivor annuities. They are of the
same extent, each requiring for its proper display a triangular
space. These are combined, therefore, as shewn in the
diagram, the joint-life annuities occupying the upper triangle
and the last-survivor annuities the lower. To distinguish the
two—although the distinction is hardly needed—the two sets
of annuities are printed in different type; and they are
separated from each other in the several columns by horizontal
lines. The pages not being of sufficient depth to receive the
whole series of ages from 10 to 96, the quadrangular space is
divided horizontally into two, of which the first, (of the three-
per-cent table,) occupies pp. 141 to 155, and the second pp.
156 to 170. To facilitate the following out of any of the

lv

columns numbers are placed at the bottom of the pages,
which in the first part, pp. 141 to 155, refer to the pages where
the remaining portions of the columns thus indicated are to
be found; and in the second part, pp. 156 to 170, the refer-
ences are to the pages where the preceding portions of the
several columns are to be found.

The directions at the bottom of the several pages of the
table, will be found, after a little practice, quite sufficient to
enable the annuity, whether joint-life or last-survivor, on any
specified combination of ages, to be readily found.

There being no room in this arrangement for the annuities
on the combinations in which the ages are equal, these are
cisposed by themselves in a separate table on page 140.

To ensure accuracy in the results, in the absence of columns
D and N for two lives, the annuity construction was conducted
in duplicate, by independent computers. Two sets of books,
properly ruled, were prepared, in each of which the terms of
the preliminary series, log vf,.,, were entered, and carefully
compared. The operation then proceeded simultaneously in
both sets, comparison being made from point to point, and
discrepancies, revealing the existence of error, traced to their
sources and removed, till the whole was completed.*

The numbers were afterwards taken out in both sets of
books, and carefully compared throughout; and it is hoped
that, as a result of the care exercised, no error of any conse-
quence will be found to have escaped detection.

It may be of use to mention that the two sets of annuity
values formed on pp. | and li, will be found in their places in
the printed table, those formed in the first specimen on pp. 169
and 170, and those formed in the second on pp. 154 and 155.

* In point of fact, owing to the care exercised in this part of the work, and the simpli-
city of the operation, very few errors were committed. In one set of the books—the first that
comes to hani—the errors committed in the logarithmic process appear to have been just
five, being on an average one error in 765 operations, It may be noted also, that the com-
puter in this case was a gentleman who, although an expert arithmetician, had no previous
practice in the use of mathematical tables.

lvi

And it will be observed that, as already intimated, the values
which occupy the rows in the formations on pp. | and li, take
their places also in rows in the printed table.

It will be well, before leaving the subject of the Joint-Life
Tables, to give a few examples of their applications.

Example 1. Find the value of an annuity on (48°36) to
be entered upon in I5 years.
(48°36) + 15 =(63°51)
The formula is,
nlFary = U" nf yt eta ytn
which in the present case becomes,
101@4o-26=0"*.15P49-362¢0-01-
In applying the formula we use the equality,

De 4
15 63 “51
v -16248-36 = Dus e L e

P. 149 Ge3-51 7 ‘0716 log 0°884886

12 Des 8918585
fe) Ds se ee col: 5: 741519
2 bs) se ee log 4854707
” lg .... col.5-067912

15.448:36 2°9350 log 0°467609
nstas-ae § 4°7300
The required deferred annuity is thus 2°9350; and the
corresponding temporary annuity, formed by subtraction,
is 4°7366.
Example 2. Find the value of an assurance of a unit
on (37°30).
The formula is,
Agy=1—(I—v) (1+azy);
which in the present case becomes,
A379 = I —(1—v) (1 + ag7-30)-
P. 145) 1+437-39 16°3553 log 1213659
238 Il—v .... 4, 2°464284
"456368 4, 1°677943
Agra9 °523632

lvii

The logarithm of the divisor for the annual premium,
(payable till the first death,) is in the process; and hence the
premium may be readily determined.

It will be unnecessary to have recourse to the foregoing pro-
cess, when Orchard’s Tables are at hand. These give at once
"52304.

Example 3. Required the value of a survivorship assur-
ance of a unit on (65) against (37).

The formula here is, (Milne, p. 184,)

AlL_=2(A,, 4+ 224),
zy $( ay t po-! Py-1 ’

and in the present case it becomes,
a
Age = 4(Ass. a7 > a)

P. 150 1+4¢5-37 8°7779 Mee 943391

238 1l—o wee 2°464284

; 25567 5 " T407675
Agys; 74433

P. 150 Geg-g7 8°0791 ~=—,,_ 0°907363

5 Pe ++ o010°017915
“ne 8:41934 log 0925278
64 EEE eee
150 Ge5-36 7°7999 — , 0°892089
3 pa .... ool, 0:003974
65-36 787160 log 0-896063
Pre Se Ee
264-37 8° I
Pes 41934
65-36 487160
Pe 34774
Acs-a7 "74433
Sum, 1°29207
Diff, "19659
Ae 64603 =half sum.
Tord 09830= ,, diff.

A6s-87 74433

viii

he difference between “92! and “86 is here added to and

Pos pws

subtracted from Ags.3;: the half sum and the half difference
are the complementary survivorship assurances, being together
equal to the joint-life assurance.

Here too Orchard’s Tables would afford material help.
By a single entry in them we should find at once Ags.37=.74433.
Notwithstanding this aid, the operation is sufficiently tedious
to render the possession of complete tables of survivorship
assurances exceedingly desirable.

The true values of the above benefits, as formed, first, by a
continuous process which will be hereafter described, and
secondly, by the use of the six-figure logarithms which arise in
the annuity operation, are ‘646010 and ‘098324. Those here
formed are consequently affected by errors of +2 and —2 in the
fifth place, respectively. These errors, which arise in the use of
the tabulated four-decimal annuities, may or may not be con-
sidered of importance. Generally speaking, if three decimals
only are used in the annuities, the errors will be of ten times
the magnitude of those that arise when four are used.

Finally, it may be mentioned, that the value of this benefit,
deferred # years, will be formed, as in other cases, by multi-
plying the value for the entire duration of (%.y) by the factor,

n Dan “yen
| U" Poy OF D. a.

This is overlooked by Milne, (p. 184,) and Jones, (p. 176,)
the expressions given by both of whom involve in their
numerical application just three times the amount of work that
is really necessary.

2. Of the Last-Survivor Tables.

These tables occupy the lower portion of the quadrangular
space of which the upper portion is occupied, as already
described, by the joint-life annuities. Those of them in which
the rate of interest is three per-cent. are on pages 140 to 170;
and from these the illustrations now to be given, of the
methods employed in the construction of the Last-Survivor
Tables, will be taken.

lix
The function under consideration is,

and the values of it have to be formed for all the different
combinations of + andy. It is symmetrical with respect to
the variables x and y: its value is not affected by the inter-
change of these; and therefore it suffices to form it for the cases
in which each value of one of the variables, say x, is combined
with all the values of the other, y, in succession, that do not
exceed it.

It would be an exceedingly onerous task, and withal very
unsatisfactory in its results, to form the required values
directly, by the addition and subtraction of the single and
joint-life annuities. But here the continuous method of con-
struction by differences, which has been already found so
effectual and satisfactory, does not fail us; and the tables
have been constructed by this method, at the cost of a com-
paratively small amount of labour, and with all needful
assurance of accuracy.

The following is a type of the arrangement adopted in the
construction. The main object in view here, as in preceding
instances, is to present in columns or in rows the results which
in the final arrangement are to occupy similar positions.

12 13 Id 15

12°12

13°12] 13°13

I4sI2] 14°13 | 14°r4
15°12] 15°13 | 15°14] 15°15

Call (z) the older of the two lives in each combination, and
(y) the younger.’ Then we have here the successive values of
x at the side, and those of y at the top; and it will be
observed that x is constant in the rows, and y in the columns.

lx

Now

Hence, if z alone vary, as in the columns, we have,

and if y alone vary, as in the rows, we have,

byz5=Aa,— Ayazy

From the first of these we get, as the formula for the
general construction in the columns,

Oey y= Gay t Nag —Agdgys 6. es (1.)

and from the second, for the construction of verification
values in the rows,

Asyti= Gay t Ady—Ayaey, ...- (2.)
Also, if # and y both vary, we have,
Ap y4z,= 4a, + Ady— Age, 5

and this gives as the formula for the construction of series
of terms descendine diagonally,

Qsy 1 y+l = Gay + AGz+ Aay—Apyaey . ~~ (3.)

Formula (3) comes into use only in the formation of the
series of initial terms.

In seeking to apply these formulze we are met by a
difficulty at the outset: while the differences are all essentially
negative, those that occur in each formula are affected with
unlike signs. This would seem to imply that in each case, to
pass from one value to the next, both an addition and a
subtraction would be necessary. But the difficulty, such as
it is, is easily surmounted. We use the differences which, in
their normal form, would give rise to arithmetical subtraction,
in their complementary form ; and thus the operation becomes
one of uniform addition.

xi

The arithmetical equivalent*® of a negative number is
formed in the same way as the mixed form of a negative
logarithm. Thus the arithmetical equivalent of—2576 is
17424. In the first form the whole of the number is
negative, while in the second the last four figures are
positive, and the prefix 1 only is negative. The two forms
being equal in value they may be used indiscriminately in
arithmetical operations. Thus, the algebraical sum of 4865923
and —2576 will be determined by either of the following
operations :—

4865923 4865923
—2576 17424
4863347 4863347

But the arithmetical equivalent is not restricted to the
above form. We may write it 17424, 197424, 1997424, and
soon. We have thus the power of bringing up the positive
portion of an arithmetical equivalent to any number of places
we please, which enables us often to dispense with the
necessity of writing the negative prefix at all, or of otherwise
particularly attending to it. Thus, to form the aforesaid
sum :—

4865923
9997424
4863347

We simply neglect the last carriage, which in such cases

is always a unit.

It must not be supposed that when, as in the case before
us, it is necessary to use a series of differences in their mixed
form, we require first to construct them in their normal form,
and thence to pass to the other. They are just as easily con-
structed in the one form as in the other; and therefore they
are to be constructed in that form only in which they are
required for use.

* This is the term applied by the late Peter Nicholson to these complementary numbers,

He appends a brief tract on Arithmetical Equivalents to the second edition of his Essay on
involution and Evolution, published in 1820.

Lxii

Inspection of the foregoing formulz will shew that it is
the differences of the single-life annuities that have to be used
in their mixed form in order that the operation shall be one
of addition throughout. The following shews the mode of
constructing the differences of the series a,, (p. 14), in both
forms :—

a, A A
se) 24°1484 — 1531 8469
II 23°9953 — 811 8189
12 23°8142 — 2039 7961
13 23°6103 —_ 2206 7794
14 23°3897 — 2316 7684
15 23°1581
* * * *

Since Aa,=@,4:—@,, and here a,,, is less than a,, the
first of the two columns marked A, which contains the
differences in their normal form, is constructed by subtracting,
arithmetically, the several lines from those above them; and the
results are consequently negative. The differences in their
mixed form, on the other hand, are constructed by subtracting,
arithmetically, each line from that below it. These occupy
here the second of the two columns marked A. The negative
prefixes, however, being omitted, as not required for our
purpose, the portions exhibited are essentially positive.

This column was completed for the purpose of the
present construction; and it will be well now to shew how
the differences of the other series, a, were formed. In-
spection of the formulz shews that these, to admit of their
combination with the others by arithmetical addition, require
to be exhibited in their normal negative form.

The differences of @,., involved in the formule are, as indi-
cated by their symbols, deduced from different successions of
the terms of that function. Thus, the difference involved in
formula (1), symbolized by A,2,.,, implying that in the primary
functionz takes the increment unity in passing from term to term,
while y remains constant, is formed from the rows in the table,
pp. 140to 170. That in formula (2), which is symbolized by

Lxiti

4,4, is formed from the columns, in which, as we know, y
varies, while x remains constant. And that in formula (3),
symbolized by A,.,4,.,, is formed from series of terms des-
cending diagonally, in which order both x and y increase by a
unit.

1. To form A,a,, It would no doubt have been quite
possible to form all the series of differences wanted from the
copy of the table written out for the printer. To have done
so, however, in the case of the present series, would obviously
have been a most irksome task. Every row has to be
differenced, the subtractions have to be performed sideways,
and there is no space whereon to set down the results. The
formation was therefore superposed upon the original com-
putations. There the rows, although in reverse order, are the
same as in the printed table; and there is ample space for
setting down the differences as formed, so as to permit
revision in case of a discrepancy shewing itself.

The method employed will be clearly understood from the
following specimen. It consists of the terminal portion of
the six (double) columns of the formation of @,,, the initial
portion of which is given on pp. I, li.

The subtractions being made sideways, the differences are
seen here ranged each under its minuend.

2. To form Aya,,. This difference, as stated, is formed
from the columns. It comes into use for the formation of
series for verification ; and as only four or five of these are
required, perhaps the best method of proceeding is to copy
out the proper columns and difference them in the usual
way.

A specimen of the main formation will be given pre-
sently. The formation of the series requisite for its verifica-
tion is here shewn.

) fe)

10
II
12
13
14
15

lxiv

Formation of agy, and Maz,

982896
339446
41

Verifications of a;,.
5 A 215
20°4046— 675 7910
"3371 920
"245I—1 Be 215-10
°1327 1250 ,
°0047—= 1386 15°10
19°8661
IS‘11
15°12
15°13
15°14
15°15

23°1581 p. 14
24°1484
47°3065
20°4046
26'9019

3?

» I4I

Ixv:

Formation of ayy and ha,., (continued).

ee fe | es 9 a, |

983828
329051
85

— eee | Oe, § eee | eee

312964) 20°54 572
152

The column headed 15, p. 141, is copied out, (the value of
@15.15 being taken from p. 140,) and differenced. On the right
@isio is formed as shewn, and the successive terms ajsn, &c.,
are formed by the continuous addition of two series of
differences, of which, the first is Aa, formed on p. Ixii, and
the second is A,@,.,, formed in the column adjoining on the
left.

3. To form A,.,@,... This difference comes into use only in
the construction of the initial terms of the several columns in
the main formation; and in these xs=y. The formula conse-
quently for this case becomes,

and the series to be differenced will be that on p. 140, headed

Qe The formation of A,.,4@,,, was, like that of A,@,,,

superposed upon the joint-life computations ; and on reference

to the specimen on p. Ixiv, the values of the former of these will
t

lxvi

be found in the first column, (in which x=y,) under those of
the latter which fall in the same column, but necessarily in
reverse order.

The following is an example of the formation of the initial
terms of the several columns :—

Initial Terms.
Qo «=: 2.41484 pp. 14
9910 24°1484 99 «99
482968 .
91010 21°007Q ,,140
IO°I0 27°2889

6938

1561 ae
rrr °1388 Verification.

6378 Qj, 23°158%

20231 9999 23°1581
12°12 26°9787 46°3162

5922 99 15°15 19°8661

2386 15°15 26°4501
13°13 “8005

5588

2646
I4'l4 6329

5368

2804

1515 “4501

Of the two series of differences here employed the first is
2Aa,, the double of the series formed on p. Ixii, and the second
is A,.-@,,, Whose formation has just been described.

The principal formation, like others that have preceded,
occupies a triangular space, having 87 columns and 87 rows.
The initial portion of the first six columns is here given.

ixvii

Formation of a5,==05+ ay — Gzg-y.

14 15
26'9787
7961
1207
26°6329
7084
1418

"5431 | 26°4501

The values of x are at the side and those of y at the
top. The initial terms were first inserted, and next the
verification series. The series Aa,, (8469, 8189, &c.) was
then entered in column 10; and the successive terms of it
were carried out horizontally, as shewn. The series A,a,-y were
then inserted, that which in the specimen (pp. L xiv, lxv,) occupies
the last vow, in the first column; that which occupies the
next to the last row, in the second column; andsoon. The ©
numbers opposite the several vows, in fact, indicate the columns
in which the differences in those rows are to be entered,
respectively. The final additions were then proceeded with,
without interruption.

The method employed in the construction of this table is
in description somewhat tedious ; but it will not be considered
so in application, by any one accustomed to the construction
and verification of tables, when it is mentioned, that the time
occupied in the construction of a complete table did not exceed

lxviii

464 hours. Of this time 24 hours were required for the for-
mation and insertion in their places of the initial and the
verification series, and the remaining 44 sufficed for the for-
mation and transference of the differences, and the final
additions.

— It may not be out of place to point out that while, as in
the case of a joint-life annuity and its corresponding assurance,
we may pass from a last-survivor annuity to a last-survivor
assurance either by the usual formula _

Asy=I—(1—2)(1 + aay),
or by the use of Orchard’s Table, a like analogy does not hold
in regard to the mode of passing from a whole life to a de-
ferred last-survivor annuity. The component annuities here,
@,+@y—a,,, must be dealt with separately.

3. Of the Construction of Survivorship Assurance Tables.

The present volume contains no Survivorship Assurance
Tables. A complete set of such tables would form a most
desirable addition to those that are here given. The Council
have no immediate intention of undertaking their construction,
what they have done, in connexion with the H™ table at
least, being sufficient to admit of the exact treatment of all
cases:in which not more than two lives are involved. The
field is therefore open for such members of the Institute, or
others, who, recognising the additional facility in the treatment
of the more complex cases that would be conferred by the
possession of a set of Survivorship Assurance Tables, have
the time and the inclination to undertake their construction.
While by so doing the benefit they would confer on their
profession would be great, that which they would derive per-
sonally from the exercise would be by no means small.

To facilitate this construction, by whomsoever it may be
undertaken, the tables on pp. 236 and 237 have been formed.
We proceed to explain the methods employed in the forma-
tion of these, and shall then shew the manner in which they
are to be used in the construction of Survivorship Assurance
Tables.

Ixix
The functions tabulated are log (f,-!— 1)and log$(p,7' + 1).
We have,

l.— aoe
L545.

log( ~,°!'—1)=log = = =log 2 =log d,+ colog /;,,.
batt

*, Alog (~.'— = =A log d, + colog ps4, ;
and hence, log (p23, — 1) =log (p,7'— 1) + A log dz + colog 5,1.
Again,

log (2 '+ 1)=log i=" pees

=log $+ log (4, + la41) + colog 4,41.

Hence,
A log (p21 + 1) =A log (4,+ 4a41) + colog p41 ;

* logd (Pepi + 1) =logh( pe" + 1) + Alog Ue +441) + colog p41.

By these formule the series were constructed, a prelimi-
nary in the case of the second of them being the formation of
log (/,+441) with its differences, a very simple matter. The
following is an example :—

Formation of log ( pz~'—1). Formation of log 4( pr +1).

Log dy 2°690196 Log 5 1698970
Col. 2, 5:002133 » (hot) 5°299965
10 3692329 Col. Ai 5-002133
908595 10 0001068
1736 998065
iI -602660 __ 1736
918405 rr 000869
1444 998409
12 522609 _ 1444
942196 12 000722
1268 998644
13. °465973 __ 1268
* * 13 ‘000634

@ #

A survivorship assurance on (x) against (y) is an assurance
payable at the end of the year in which the combination (xy)
is dissolved, provided the dissolution is caused by the death
of (x). When the sum assured is a unit the value of the
assurance is denoted by Aj;

The value of this assurance may be conceived to be com-
posed of two portions, one having reference to the first year,

; xx

and the other to all the years after the first. Now it is shewn,
(Gray's Tables and Formula, page 73,) that the probability of ;
(x) dying before (y) in the first year is

4(1—Pe)(1+fy) ;

hence the value of the first portion of the assurance is,

40(1—22)(1 +f) ;
and the value of the second portion is obviously,

i
UoyAz- l-y+ 9

in which A=j>7; denotes the value of a similar assurance on
(7+ 1) against (y+1).
Hence,
Az, =8(1 —pz)(1+f,) —peyAsignl
=Upeylt( he '—I)( 2+) + Asari ;
or, in logarithms, .
log AZ, = log Ur, + log [4$(2.7'—1)(4,71 +1) + Asami:

This is the working formula, by means of which, commenc-
ing with a combination in which x and y, having between them
a specified difference, one or other, (or both,) is the greatest
tabular age, we may form in succession, the values correspond-
ing to all the combinations in which the variables have the
same difference. For this purpose, and for the formation of a
complete table, we require, first, log uf,., for all the different
combinations of # and y. These logarithms have been already
formed for use in the construction of a,.,, and would of course be
immediately available. And we want, secondly, to be used as
will be presently shewn, log 4(,-'—1)(f,'+ 1), not for the
different combinations merely, but for a// the combinations of
the two variables x and y. The cause of this duplication of
the number of logarithms to be formed—for it amounts to a
duplication—is that this function is not, like the other, symmet-
rical with respect to x and y.

The simplest arrangement for the formation of log
3(p2-'—1)(f,7'+ 1) would be in a quadrangular space, consist-
ing of 87 columns and the same number of rows, having the

Ixxi

successive values of, say, z, at the top, and those of y at the side.
But this arrangement is objectionable inasmuch as the series
in which «—y and y—# are constant, (which are the series
more particularly required,) would find their places in it in
diagonal lines; and thus a difficulty would be interposed to
the taking of them out in order. The formation will there-
fore be effected in two compartments, of both of which the
following is a type :—

In the first compartment +>y; and, the successive values of
x being at the top, those of +—y are at the side. In the
second compartment +<y; hence, the values of y being at the
top, those of y—-+ are at the side.

1. *>y. In descending diagonally x varies in reverse
order. Hence the initial terms will be formed by the con-
tinuous addition of the negative differences of log (~,~'—1),
in reverse order.

In the rows both x and y vary, in reverse order. Hence
verification series will be formed by the continuous addition
of the two series of negative differences in the table, in reverse
order.

In. the columns y varies, in direct order. Hence the prin-
cipal formation will consist in the continuous addition of the
terms of the series A log 3(f,-'+ 1).

2. x<y. Between this case and the last there is simply
an interchange of the functions in x and y. The modification
this causes in the working formule is easily seen; and it will
receive illustration in the examples now to be given.

It is necessary first to form the initial and verification
series. The following are specimens of these formations :—

Initial Terms.

i +1) 0-001068

Leg, (rai 1) 0-647817
9 4

96°10 0°648885

596485

"10 «6 °2.45370
9s 768379
94°10 °013749
839603

93°10 © 853352
881377

92°10 6734729
924397

giro §=—°659126

* *
Initial Terms.

Log 4(pal+-1) 0-508155

?

(pio! 1) 3°692329

g6"10 9-200484

95°10
94°10
93°10
92°10

gI°1o
*

765437

‘965921

906767

Ixxii

a>y

z—y=81

Log (px!

» 4(pia!+1) 0-000625
9615 0-648442

95°14
94°13
93°12
92°11

QgI°1o

<Y

596485
999976
‘244903
768379
33
013315
839603
88
"853006
881377
147
"734530
924397
199
"659126

y—2=81
‘| Log 4(p.g'+1) 0°508155 Log 4(pxy'+1) 0:508155
» (py'—1) 0647817
96°96 1:155972

» (prt—1) 3°459280

96°15 3967435

95°14

94°13

93°12

Q2°11

gI‘ro

765437
983071
716943
906767
023622
"646332
951849
056536
"654717
971865
080151
706733
984957
089669

781359

Verification Series.
z—y=0
—1) 0°647817 Log (pg!—1) 0:647817

» $(pa't+1) 0508155
96°96 1:155972

95°95
94°94
93°93
92°92

gr°g!
*

Verification Series.
¥ -——7==0

95°95
94°94
93°93
92°92

Org!
*

596485
765437

*617894

768379
906767

"193040

839603
951849

"984492

881377

765437
596485

"517894

906767
168379

‘193040

951849
839603

"984492

971865

IXxiti

The following are specimens of the formation of the principal
series :—

|

36

85
84
83
82

81

82

81 |

Formation of Log} (pe'—1 (py! +1).

96
648885
999801

95

r>y.

94

93

Q2

oY — ——— el en ed wwe fo See ee

648686
999853

ee eee

648539
999912

245370
999801

eee

245171
999853

013749
999801

———ee— ff fee fee fo Oe

648451
999967

245024
999912

013550
999853

853352
999801

one =f nee ef = Oo eee

| 648442

736847

999967
244903

‘321276

013403
999912

013315

080689
8966

853153
999853

853006

915726
4566

734729
999801

734530

5

794506

Cane)
oak | Sesame § oe Se

333332
15043

659126 |

715246 |
3657

a, a ee ee ee

780025
48151

828176
93233

921409
234563

155972

g6 |
200484
910331

110815
919849

030664
943464

974128
976378

950506
16929

967435

348375
28135

376510
48151

424661

95

a)

965921
910331

876252
919849

796101
943464

739565
976378

715943

&

101711
15043

—_—

116754
28135

144889
48151

193040

t<y
94

929258
12056

941314
15043

956357
28135

984492

93

801669
8966

810635
12056

822691
15043

837734

g2

721500 |
4566 |

726066 |
8966 |

735032
12056 |

747088 |

gI

eee OR eee

eee § eee fee 0 eee

872688
910331

783019
919849

702868
943464

' 646332

824537
910331

—_—_ ——- —

734868
919849

ee ee

654717

796402
910331

106733

781359

k

Ixxiv_

96 95 94 93 92 QI
5 | 166213 | 862236 | 710611 | 629936 | 582319 | 538484
75603 | 69414 | 58392 | 32524 | 19482 | 28792

pe eee ee Oe ae

4 | 241816 | 931650 | 769003 | 662460 | 601801 | 567276
118623 75603 69414 58392 32524 19482

3 | 360439 | 007253 | 838417 | 720852 | 634325 | 586758
160397 | 118623 75603 "| 69414 58392 32524

ne re EE eel

2 | 520836 | 125876 | 914020 | 790266 | 692717 | 619282
231621 | 160397 | 118623 75603 69414 | 58392

1 | 752457 | 286273 | 032643 | 865869 | 762131 | 677674
403515 |.231621 | 160397 | 118623 75603 69414

pS NE minnie erm neem ce, TI TO el

o | 155972 | 517894 | 193040 | 984492 | 837734 | 747088

It may be mentioned in regard to the preceding formations,
that when the initial and verification series have been inserted
in their places, the whole of the subsequent work need not, and
will not, occupy a fairly expert arithmetician more than
twenty-four hours.

Also, since the element of interest is not involved in these
series, when they have been once formed, they can be used in
connexion with any rate of interest that may be proposed.

Having now got the auxiliary series, we are prepared for
the construction of the required tables of Az. .

The formula is,

log AZ,=log up, + log[$ (.°'—1)( py! + 1) + Anza,
which, for the moment, we will write, .
log a=log 6+ log (¢+ 2).

Here log 4, log c and log d* are known; and to obtain

log a@ it is requisite first to form log (¢+¢@).

This of course can be done by the use of the common
tables, but the operation would be a tedious and laborious
one. Fortunately this labour need not be incurred ; the end
in view can be attained with great facility by the aid of the

* It will be presently shewn that when either x or y is the oldest tabular age,
log Asisty that is log d, is known; and from tius value we descend to those corres-

ponding to the younger ages, in succession.

lxxv

table employed in the formation of @,,. The table in question,
as already mentioned,* gives log (1+ 2) corresponding to suc-
cessive values of log; its characteristic equation being,

T [log z]=log (1+),
in which T is a functional symbol denoting the tabular result
corresponding to the appended argument.
a

In this equation write = for z, and we get,

T [log ss or T [log d—log ¢]
_ a c+,
=log (1+ ¢ )=log(——) =log (c+ d) —logc.

. log (c+d@)=log c+ T [log d—log c].
It thus appears that log(c+d) is formed by adding to
log c the tabular result corresponding to log d—log c.
We have, therefore,
log a=log 6+ log c+ T [log d—log e] ;
and, changing the order of the first two terms, the work is
very commodiously arranged as follows :—

: log d

loge log d—log c
log 4
T [logd—log c]

|
|
|
|
| Sum | log a

Log d is supposed to have been just formed; and, log ¢ and log
5 having been previously inserted in their places, loge is sub-
tracted from log d, and the tabular result answering to the
difference is written under log 6. Addition of the three lines
then gives log a, which forms the log d, of the next: operation.

The type just given is here repeated, with the values of
a, 6, c and d restored; and the arithmetical operation by
which we pass from log A>, to log Ab is clearly exhibited.
The numbers corresponding to the logarithms successively

formed are not to be taken out till the close of the operation.

* P. li.

lexvi

1 i
log Asisz | Aisne

| Ds
A | log4(p.—1)(f,14+1) | DA
B ” Pay
C
pf

An

The arrangement adopted in the formation of Aj, is
similar to that employed i in the construction of @,,. It is
typified as follows :—

4. § 6

| |
|
|
{
97°93
96°92 | 97°92
95°91! 96°91 | 97°91
¢ | e ¢

Here for +>y, «—y, (constant in the columns,) is at the
top, and y, (constant in the rows,) is at the side; and
for +<y, y—+ is at the top, and x at the side. This arrange-
ment affords the greatest facility for the transfer of the
values formed to the tables arranged for use; since in these
the values that are here found in the rows, will take their
places in either columns or rows.

The first thing now to be done is to form the initial values
for the several columns.

The expression in its first form is,

A =0[R(1 —h.) (1 +4) + hey Aszizail:
Case 1. x=y. Let x and y each equal 97. Then £, fA,
and f,., vanish, and the expression reduces to,

A sy = to ;
whence,
log Aga, =log *5 + log v.

Ixxvil

Case 2. +>y. Let r=97; then 2, and f, vanish, and we

have,

A=! +h) = Upy (py +1);

whence,

log Aj, =log up, + log 4( py" + 1).
Case 3. x<y. If y here equal 97, p, and f,, vanish, and

we get,

Ajg=4tr(t ps) =4Uhs(P2'—1) ;

from which,

log Ai; =log ‘5 + log up. + log (ps7

For Case 1 we have,

log*5 1-698970
» 2 1987163
1686133

For Cases 2 and 3 the initial terms may be formed in series,

as follows :—

Initial Terma.

a>Y
Log tpye 1:251209
» (pos! +1) 0°508155
97°96 1-759364
295816
765437
97°95 ‘820617
132721
906767
97°94 860105
73995
951849
97°93 885949
45456
971865
97°92 °903270

9791 913316

® e

z<y
Log °5
x Upge
”? (pog7! — 1)

96°97

95°97

94°97

93°97

92°97

91°97
e

1:698970
1:251209

0:647817

1597996

295816
596485
490297
132721
768379
391397
73995
839603
304995

Ixxviii

These series being, almost necessarily, formed in reverse
order, it is the complementary differences of their component
functions which constitute the addends. Thus for +>y the
addends are, A colog uf, and A colog 4 (f,-'+ 1); and for r<y
they are, Acolog vf, and A colog (g,-'—1). For Acolog #,
and A colog uf,, (the series being identical,) we use A colog fy,
as on p. xliv, subjected to the corrections there specified ; and
the differences of colog }(f,-'+ 1) and colog (g,-'—1) are the
negative differences of log 3(¢,-'+1) and log(g,-'—1). They
are of course used in reverse order.

The specimens of the main formation will be found on
pp. lxxx, Ixxxi, lIxxxii and Ixxxiii.

The initial terms are first inserted in their places;
and then, in accordance with the type on p. Ixxvi, the
terms of the two auxiliary-series, which are taken, the first
from the formation on p. Ixxiii, and the second from that
on p. xxiv.

The working then commences, and is carried on in each
column, till all the columns in succession, in both compartments,
are brought down to the same point, say y=85, and +=85,
when measures will be taken for verification, as will be
presently explained. Referring to the column r—y=0, (2 >),
155972 is subtracted from the initial term, which is log Aig,
and the difference 530161 is set down opposite the subtrahend.
It is easily ascertained that the index of this logarithm is 2.
Entering the table then with 2°530161, the result, 014477, is
set down as shewn ; and summation of the three lines gives
for log Aim, 685705. This forms the initial term for another
similar operation, the result of which is ‘684407=log Ag.
And so by a succession of such operations the column is
completed, and‘ all the values of log A; in which e=y are
formed. The operations in the other columns also, are entirely
analogous to that described, the values formed in each being
those in which the differences between 2 and y are as at the
top, respectively.

The same remarks apply to the second compartment, in

lxxix

which z<y. The first column here would be identical with the
corresponding column in the first compartment, and therefore
it is unnecessary to insert the figures.

It is, as has been said, the remaindets of the subtractions
that occur in the several operations, that form the tabular
arguments. These regularly increase in magnitude, and it is
therefore sufficient formally to determine the index of the first
of them, since, in succeeding remainders an increase (of a unit)
in the index is always pointed out by a decrease in the mantissa.
Examples are seen in the first three columns of both compart-
ments, where, after the first step, the index increases from 2
to 1, the increase in each case being accompanied by a
decrease in the mantissa.

id

For the verification of the foregoing construction, it is re-
commended that the work be performed in duplicate by
independent computers. It ought to be carried on in sections,
each embracing twelve or fifteen of the ages at the side, and
ending in both sects of computations with the same row.
Comparison of this row in the two sets will show if error has
been committed; and if so it must be sought out and
removed. Each section must be finished and corrected ere
commencing the next.

But if, as will usually be the case, the annuities on the
joint-lives have been previously formed, verification can be
otherwise obtained. For we can, by Orchard’s Table, pass
from the annuities to the corresponding assurances ; and we
can also, by addition of the proper values in the table being
formed, find the values of the same assurances. This follows
from the equation,

Azy=A5+ A=;

zy?

that is, the sum of the two survivorship assurances in which .
the ages are interchanged, is equal to the assurance on the
joint-lives. And the values in which the ages are thus
interchanged are very readily found, as they occupy corres-

97

96

95 | 68

9+ | 682641

93 |

g2

Ixxx

Formation of At,

&>y

685705| 484959] 759864) -574598

738103:

837734! -842895
611232 565776
| 229604 205454

39 17!

661408 636319
268083 256078
16 6!

681949
291145
30

{ -577298| 820617

: 77604 "534078
822691! 90491

661633

01 376510 '1-436169
313603
meer

547140] 794940! 623651
956357) °7817461116754' -678188

491781)
169245'
29
777809! "599527
941314) -836495
590863
226888

676589] -47488¢| 715004 518912 750i 7574254
735039| 980062 810635} 948469

656860
275996
32

697492
309817
29

733404] °541257

94

93

Q2

gi

go

89

Ixxxi

Formation of Az, (continued).
9 >y

y—T=3 ¥—xr=4

|

/

y—2=5

870575| *7422092| 903270! -800332

101711| -724755| 333332| -537243| 736847| 1-166423

384147
128564

611404 537409 404688
242626 196054 135402
34 32 14

ee ee |) en eee add

783322| -607186| 823160) -665516| 861380

801669) 9816531 920292] -9028681 080689

672403 626947 552952
291927 255144 205040
26 30 35

—ee | ee ee | ff er | nen

766026) -58347¢] 802413) 634473] 838716

* * * * *

r<Y
x ! y~r=1 | Y—z--2
97 ee
96 484959 |597996 “39674
752457. 2-845539,
811071
029377
35,
95 592940; ~ "391688 490297, 309241
286273) 1 306667, 520836: 9-969461
239608 943792!
080115 038664
11, 41
94 606007 +403 562, 603333) .318664
032643) -673364' 125876, 1377457
446324 313603)
188109 092879,
, ll
93 "4.79323 STORE 908 "414084! 539369! -340606 "340098
865869, 751219 914020, -618349
565776 491781
194208 150830
7 15
92 °4$76990 | 625860! -422532) 556646 _*360285
762131| 863729, 790266| -766380
590863
199714
29
91 | "474886 '433179| 580872) -380954
°958993)| 692717, -888155
681949 656860
280966 24867 4
45 24
go | 492777 640634 "437 154) 59827 598275 "396529
607226} -033408 619282) -978993
718031 697492
318051 290608
4 ; 46
89 "470697 | 643312) °43¢858) 607428) -404975

Ixxxiii

—

y—2=4 y—ar=5

i

er ee me

——— Ee ee eee | eee | eee |

SS

O4 391397 _'2.46262)
360439, 1-030958!

oO

93 | 422526) -264567, 304995; *201834
415273) 241816) 1063179)
063244
047523
es SS
"292917| 352592) :225212| 231828
628328) 931650
384147
101600
9
322910) 417406) - °200500
788229; 769003) - 440093
537409
159880
1
340883) 466293 "235830
905858) 662460) - 661987
626947

*170541
1065615

256479 213917
26 _j|__18 see
8, |563233| -365791| 805337| -318607| 427673) 267715

e * @ e ® e

‘Ixxxiv

ponding positions in the two compartments of the formation
just given. The joint-life annuities also, with which we have
to do, occupy positions corresponding to the others in the for-
mation on pp. |, li.

Thus, to verify the rows opposite 92 :—

92°92 92°93. 92°04) (92°95 s«Q92'96.—s-«9)2'97
476990 534078 599527 670603 742292 800332
476990 422532 360285 292917 225212 170541

— -eee

"953980 ‘956610 ‘959812 "963520 967504 °970873
The annuity values are :-—
"5800 «6. 4898S 37908 )=— 2524 = II57 = 0000

and Orchard’s Table gives for the corresponding assurances :—

95398 ‘95661 ‘95982 ‘96352 ‘96751 ‘97087

This will no doubt be considered a sufficiently close cor-
respondence between the two sets of values. The two forma-
tions will, however, stand a yet severer test. If we deter-
mine the assurances from the annuities by the formula,

A,y=I—(I—v)(I + az),
(log (1 +4,y) being found in the annuity formation,) we obtain,

0:198668 0°173120 0°139815 0-097745 0-047538 0-000000
9464284 2:464284 9-464284 9464284 2.464284 9-464284

SES ES eR es

2662952 2°637404 2:604099 2:562029 2-511822 2-464284
"046021 7043391 ‘040188 . 036478 = "032.495 "029126
953979 . 956609 ‘959812 963522 = 967505 = -9 70874
and we have now an all but perfect agreement in the sixth
place.
This method of verification will be available when the
work is undertaken by a single computer; but it may also be
occasionally had recourse to when the work is done in dupli-

cate.

The final arrangement of the results will be analogous to
that of the annuity values, pp. 141, &c.; and they will occupy
the entire quadrangularspace. The values in which x=y will
take their places in the diagonal of the quadrilateral; and, if

Ixxxv

the values of # be put at the top and those of y at the side,
the values of the function which occupy the rows in the first
compartment of the formation will here take their places, a/so
i rows, above the diagonal, while those that occupy the rows
in the second compartment, will take their places 7% columns
below the diagonal.

This will be understood by reference to the following
specimen of the final arrangement :—

Values of Assurance of a Unit, on (z) against (y).
(z at the top, y at the side.)

a
* % % 2

#

+ * | * * | |

| 89 | °58248| 63447 ors | 89
| 90 | °55402| "60719 66551) °72674) go |
1 | °51891 57425) 63567, ‘79154 °76670 T |
| 2 | *47699] °53408, 59953. 67060] 74229] ‘80033 2

| 3 | 42253) 47932) “54715, 62365, °70328| 76904) 3
| 4 | 36029] *4.1408) °48155 56184 64965| "72461, 4°
95 | °29292| -34070} 40365! -48351, 57730, 66163] 95 |
6 | '22521} 26456] °31866) -39160) -48496) °57460 |

7 | *£7054) '20183] 24626) -30924! *39627] 48544) 7

The specimen forms the concluding portion of the last six
columns. It is composed of the values formed in the examples
on pp. [xxx to Ixxxiii.

Enough has now been done, in regard to the function
under consideration, to enable any one, otherwise qualified for
the task, easily to complete the three per-cent. formation, or
to undertake the like at either of the other rates of three-and-
a-half and four per-cent.

It does not seem too much to expect that among the
younger members of the profession, who hope yet to occupy
higher positions in it, there are to be found some who, recog-
nising the claim their profession has upon them, as so well set
forth in the passage from Bacon which forms the motto of
the Fournal,* are willing to devote a portion of their spare

* ¢¢] hold every man a debtor to his profession, from the which as men of course do seek

to receive countenance and profit, so ought they of duty to endeavour themselves by way of
mends to be a help and ornament thereunto.”

Ixxxvi

time to the extension of the deductions from the tables which
have, at the cost of so much time, labour, and expense, been
prepared and put forth by the Institute. Such a course would
be remunerative in various ways :—First, it would beget the
satisfaction arising from the discharge of a duty which, by
hypothesis, they feel to be incumbent upon them. Secondly, it
would lighten and facilitate their labours when they come to
occupy the positions to which they aspire, by increasing the
stock of material required for the proper and satisfactory
conduct of their business; and, thirdly, it would eperate to
their more immediate advantage by imparting to them a
dexterity in manipulation and a confidence in the use of tables
not easily acquired by other means.

The present Introduction has grown to what it is feared
will be considered by many, an undue length. The object
mainly kept in view in its preparation has been, to show with
what facility, by proper arrangement, and with the aid of the
materials contained in the present volume, the tables required
for actuarial use may be constructed: it is hoped that it will be
found to contain little which does not bear more or less closely
on that object.

P, G.

EXPLANATION

OF THE

ADJUSTMENT OF THE TABLES.

BY

W. 8S. B. WOOLHOUSE.

Tue Experience Committee having intimated a desire that I should
adjust the graduation of the Tables of Mortality for Healthy Lives, the
task undertaken by’ me is completed, and, for the information of the
Committee, it only remains to accompany the resulting Tables by an
explanation of the details of what has been done. The subject being
one of very great practical importance, inasmuch as the adjusted tables
are designed to form the basis of an extensive superstructure of other
tables, I have given to it a carcful and independent consideration, and,
after testing various schemes, I have at length succeeded in devising a
method that may be regarded as efficient and satisfactory.

AS suggested in my paper® “On the Construction of Tables of
Mortality,” the number-living at each age is the most manageable element
for final adjustment; at ages beyond the limiting age of the Table it is
at once conveniently, as well as accurately, put down as zero, a practical
facility that cannot be over-estimated; and it has this essential advan-

* Fournal, vol. xiii., p. 95.

Ixxxvili

tage, that precisely the same aggregate tabular mortality or decrement
must necessarily be retained between all points of actual coincidence,
in whatever way the intermediate numbers may be modified. As a
consequence of this principle the number of such coincidences with
experience in the curve of the number-living may be regarded as one
test of close adjustment and substantial exhibition of the actual mortality.
The method I have adopted may be briefly stated, and the rationale of
the process and its accurate adaptation to what is chiefly required will
be at once apparent.

The data, for the reasons already stated, are the numbers-living at
successive years of age as deduced, without any adjustment, from the
original facts. If we begin at the first age in the Table and extract the
numbers-living at quinquennial intervals, that is, according to the usual
notation, /,9, /15» /es 445, .... we can by the formula for interpolation
determine all the intermediate values at the other ages, and so obtain a
complete series of values that shall be continuous. Geometrically
speaking, we shall thus pass a continuous curve-line through the indicated
quinquennial points. Against the adoption of such curve-line as the
basis of the final Table there is manifestly this tangible objection, that
the numbers at the ages 10, 15, 20, 25, .... are made use of exclusively,
and that the original numbers between these ages are wholly ignored as
data. This rather material objection, which is inherent in other methods
of adjustment, is entirely removed by varying the epoch of the adopted
quinguennial data, that is, by taking the five distinct series hereunder
stated, viz. :—

lio» Ass hi, Lys, eves
Ar» hie» fas he, cee
4a» Ar has fers cee
hs, Ags hs» he, cere
Nias hos fea» Mop, ++

then by separately interpolating the intermediate values for each of these
series, and by finally taking the arithmetical average, or mean value, of the
five completed sets of results, the series of adjusted values is obtained.

Reverting again to a graphical illustration, all the points of the
original data are thus occupied by five distinct curves, assimilating to the
experience and to one another, and forming in combination a sort of
network; and at every age the resulting ordinate of the adjusted curve is
the arithmetical mean of the five corresponding ordinates, and, in other

Ixxx1X

words, the five curves are, as it were, mutually drawn in towards a central
course. ‘That such central curve must exhibit a correct average of the
original observations without giving unduc weight to any of them,
it is unnecessary to explain, and it will be at once perceived that every
element of the data is equally employed in its determination.

It is not requisite, however, to compute these five curves separately,
a labour which would be unnecessarily circuitous, For the purpose of
actual calculation we proceed mathematically to reduce the preceding
system of operations to a direct process.

For any given age let / denote an interpolated value of the number
living, and let /,, which denotes the original number for an age z ycars
older, be the nearest quinquennial point of the corresponding curve.
Then the series of values from which / is found are /,_,, 4, 445, &c.,
and by interpolating with central values and stopping after second-differ-
ences, we shall have

ie
=a (na

SH), 4 ot, XS)

For the several values of /as deduced from the five respective curves
we must make z separately equal to —2,—1, 0,1, 2. The five values
of / are therefore

== —°121_7+ °84/_5+°28/,3
d= — *08/_ 6+ °96_1 + 12044
f= o+ 2+ Oo
f= +1 2l_a+ "96/41—"08/,,
d= + 2873+ °84/,9—*1 2/47
Hence if we put
N=lLitles, ye=lat lye, ete . )
YeHLatla, ve=lot lee, y= tr
fen 8=1—- Ys h=yom ys A= + MP)
and if (/) denote the required average number-living at the given age, wc,

shall have by adding together the five values of /,
m

xc

S(Q=l+'96y + B4yo+'28 73+ 12 y4—"O8yg— "I 2yz
=l+ yi + y2—'04{ (v1 — y3) + 4(va— 73) + 2(¥e— 3) + 3(77 4) }

=l+ yit72—04(/+ 4g + 24+ 3h) .

(y)

By means of these last formula, (a), ((3), (y), the required adjusted
values of the numbers-living are readily computed.

Example. In the Table H™, Healthy Lives—Male, it is required to
find the adjusted value of / for age 25.

The data are taken from “The Mortality Experience,” Table H™,
page 273, and the calculation sz extenso is annexed :—

1 2 3 4 | G | 7
a a
t 9297 | 7, 9861 | ts 9434 | 1-3 9493 | 24 9860 | /¢ 9684 | 1.7 9743
yi 18610 | 241 9249 | 242 9185 | Uys 9125 | Lys 9054 | lye 8913 | l47 8848
y2 18619 | 1 18610 | 318619 | ys 18618 | y4 18614 | yo 18597 | y7 18591
| 46526 |_ys 18618 | ys 18618 | 24 - 42 -ys.18618 | 7, 18614
Cor®+460| f -8| g +1| 3k - 69 h -21/k -23
5)46530°60 | 4g +4 -l
() 93061 <4] ot. - 4
-115 | x - 04 | = 44:60 | Cor

In filling in the data the computer will bear in mind that a chasm
precedes the last two columns, and that /_, and /,, are there passed over
unheeded. This is further indicated by the numerals placed at the top of the
respective columns. It will be further observed that the fina! result is
here put down and retained to an extra place of figures.

Since the expression (y) is linear with respect to the several values
of /, it is evident that precisely the same formule may be applied to
adjust the yearly decrements, and these being much smaller numbers the
calculation of the table may be thus considerably abbreviated and
expedited, the numbers-living being then deduced by the successive
subtraction of the adjusted decrements. ‘The adjustment of the decre-
ments undoubtedly offers the greatest possible facility.

As an example, taking the same Table as before, the following is the
calculation of the adjusted-decrement (¢@) at age 25:—

(25)
d 48 d_; 64
yi 128 | dy, 64
183 [y RE

309 | y3 130

5)308-48) 4g + 12
(d) 617 +10
93061 |Dage25
92444 [() ,, 26

The resulting adjusted decrement 617 being here subtracted from 93061,
the adjusted number-living at age 25, we obtain 92444 for the adjusted
number-living at age 26.

In the actual construction of the Tables I have first made
independent computations of the adjusted numbers-living for every fifth
year of age, and afterwards calculated the values throughout for every
age by adjusting the decrements, thence deducing by successive sub-
traction the numbers-living, and making use of the former calculations
at each quinquennial stage, as a periodical check on the accuracy of the
work. The final results, on being differenced to second-differences, are
generally found to be remarkable for their orderly progression, though
at exceptional places there may yet exist some slight traces of irregu-
larity, but they are quite isolated, and so minute as to be readily amended
by inspection.* ‘The simplicity and efficiency of the manipulation will be
duly appreciated by those who may hereafter have occasion to put it in
practice.

Another matter incidental to the completion of the Table it will be
requisite to explain. As the data for each separate calculation must
extend over an interval of seven years preceding and following the age,
the formule will obviously not apply to the first seven years of the
Table, and the numbers for those years, viz., ages 10 to 16, will
therefore be wanting. To effect a continuous junction at age 17, I
have considered it most expedient to supply the required numbers by
means of constant third-differences. At age 10, the radix of the table
is /g=z100000. If A,, Ag, Ag be the differences immediately following
age 17, and 1==7, we shall have

* Those who are not practically familiar with progressions of differences, and the dis-
turbances caused by isolated errors, need only have recourse to the elementary Rule given
by me in a former paper, Fournal, vol. xii, page 140.

XCli

yo 4yy—2A, + —— Het Da, — at et) wet 2) A;
hy 7h-+28A4— 84A3

from which

When the series of numbers is put down in a retrograde order, the
differences that are of an odd order will change sign. In such case
therefore we shall have to begin with 4,, and apply the three orders
of differences after having reversed the signs of the first and third.
The third-difference should be calculated to one or perhaps two
additional places of figures, and then the continued summation of the
differences will sufficiently check the accuracy of the computation.

The first calculated numbers of the H™ Table, beginning at age
17, with the accompanying differences, are

Age17 = 97624

18 97245 _

“19 = 96779

Here we have A, =—379, Ag=— 87, and

—379
466 °7

_ _ 87 , 379 2376 _

3 +2 84 — 25°70
Hence, changing the signs of the odd orders, the three commencing
differences are -+379°0, —87°0, +25°7, and the retrograde calcula-

tion is as follows:—

+25°7 | =As
= 87°

I §? ‘ | +379°0 mee

7 | 97624 319-7 1°3

16 | 97941°7 go: | 356

15 | 98223°8 272'2 | ~ 99

14 | 984960 2880 «| +15'8

13 98784'0 +41°5

12 | ggtizs | 3795 | 467-2
11 | ggsr02z | ,38°7 | +929
10 | 99999°8 | T4°9
The column of second-differences is formed by the repeated addition of

the constant third-difference placed above it, and the other columns are
hence obtained by continued addition, or the inverse operation to that
of differencing.

VClil

In conclusion it may be observed that in the method of adjustment
here laid down and adopted, every individual element of the data supplied
by the experience has its proper and legitimate influence in determining the
several results, and that there does not exist anything of an arbitrary
nature in the process, * No extraneous condition or restriction is placed
upon the quantities, which are freely permitted to manifest and assert
their own law. In fact, if the original points, taken in groups, were to
range in curves of the third order, they would in such case not be
subject to any alteration whatever by the operation of adjustment.
Thus the process, unlike othcr methods of adjustment, does not in any
way interfere with the organic relations which exist amongst the true
values but operates exclusively upon the incidental imperfections, whether
in excess or defect, that is to say, its efficiency is wholly directed towards
the neutralization of the small positive and negative portions of the data
which constitute the errors of observation. Hence, also, if there should
be any particular phases in the absolute law of mortality or any special
peculiarities at certain periods of life, the same will be brought out with
greater clearness and significance in the final Table, after the casual irregu-
larities have been eliminated.

The adjustments have been separately made on the H™, H¥ and
H™? Tables; and also on an H™*) Table, excluding the first five years
of assurance, this last Table being designed for the general purposes of
valuations.

“
Oo

=

ed

we

pb © &>
WH PHONO CONF ON RW BH OW ONT AND & BH OLD ONT OM BD OM} DT ON bw BD

as

tun

I00 000

99 510
99 113
98 784
98 496
98 224
97 942
97 624
97 245
96779
96 223
95 O14
94971
94 321
93 683
93 061
92 444
g1 826
QI 192
90 538
89 865
89 171
88 465
87 748
87 021
86 281
85 524
84745
83 943
83 122
82 284
81 436
80 582
719737
78 830
77919
76 969
75973
74932
73 850
72 726
71 566
7° 373
69 138
67 852

Detre-

ment.
dy

1339

XCIV

H*®.,
_ Healthy La Lives.—Male. —Adjusted Table.

Log /, |

5°000 000 0

4°997 866 7
4°996 1306
4°994 686 6
4°993 418 6
4°992 217 6
4°990 969 0
4989 556 6
4°987 867 3
4°985 781 1
4°983 2789
4°980 521 5
4977 591 9
4°974 608 4
4'971 6608

4°968 7677

4°965 8787

4°962 9657
+9599567
4°956 8309
4°953 590 6
4°950 223 6
4°946 775 5
4'943 237 2
4°939 624 1
4°935 9152
4°932 088 o
4°928 1141

4°923 984 5

4°919 7160

4°915 315 4
4°910 8164
4°906 2380
4°9OI 5509
4°896 691 5
4°891 643 4
4°886 315 8
4°880 659 3
4°874 667 3
4°868 350 5
4861 689 7
4°854 706 7
4°847 4061
4°839 7168
4°831 5627

Log pz
=A logl,

9°997 8667
9°998 263 9
9°998 5560
9°998 7320
9°998 7990
9°998 7514
9°998 587 6
9°998 3107
9°997 9138
9°997 497 8
9°997 242 6
9°997 059 5
9°997 017 4
9°997 0524
9°997 1069
9°9907 Irz1o
9°997 087 0
9°996 9910
9°996 874 2
9°996 7597
9°996 633 0
9°996 5479
9°996 465 7
9°996 386 9
9°996 2911
9'996 1728
9°996 026 1
9°995 8704
9°995 7315
9°995 599 4
9°995 5010

9°995 4216 |"

9°995 3129

9°995 1406 ]°

9°994 9519
9994 672 4
9°994 343 5
9°994 008 o

9°993 683 2 |°
9°993 339 21°
9°993 0170 |°
9°992 699 4 | °

9°992 3107
9°991 845 9
9°991 343 8

ee
« id e e e e

Prob. Sur-

Prob. Dying

viving 1 Year.) in 1 Year.

Pz

"995 1000
"996 O10 5
996 680 6
"997 084 5
"997 238 5
"997 1290
"996 7532
996 1178
995 208 o
9942550
"993 6710
993 2750
993 1558
9903 2359
"993 3606
"993 359 9
"9935 3149
"993 095 6
"992 828 3
"992 5667
"992 277 3
"992 082 6
"991 895 1
"OOI 7149
"991 496 3
"QOI 226 3
"990 891 4
"990 536 3
"990 2196
"989 918 4
"989 694 2

"981 399 5
"980 265 9

Yr

004 900 0
"003 989 5
"003 319 4
"002 915 5
"002 761 5/
"002 8710
"003 2468
‘003 882 2
"004 7920
005 7450
006 329 0
"006 7250
"006 844 2
"006 764 1
006 639 4
"006 6301
006 685 1
"006 904 4
‘oo7 1717
"007 433 3
"007 7227
"0079174
"008 1049
"008 285 1
008 503 7
"008 773 7
"009 108 6
"009 463 7
"009 780 4
"010 081 6
"O10 3058
"010 486 8
"O10 734.4
‘OIL 1269
"OIl 556 5
"O12 192 1
"012 940 3
"013 7022
"O14 439 8
"015 2200
7 |°015 950 3
"016 669 g

“O17 549 3
‘018 600 5

po 7341

on oe

No. | Decre-
Living. | ment.

z£ &

Age.
z

|
I
66 513} 1399
65 114] 1462
63 652| 1527
62 125] 1592
60 533 | 1667
58 866| 1747
57 119] 1830
55 289] 1915
53 374) 2001
51 373 | 2076
49 297) 2141
47 156| 2196
44 960 | 2243
42717 | 2274
, 49 443 | 2319
38 124| 2371
35 753 | 2433
33 320] 2407
30 823} 2554
28 269 | 2578
25 691 | 2527
(23 164] 2464
20 700 | 237-4
18 326| 2258
| 16 068 2138
| 13.930] 2015
11915] 1883
10 032| 1719
1545
1346
1138
941
773
615
495
408
329
254
195
139
135] 86
40

i
WO Cr Qh

>.)

© WAT be wn 8

ha |

~“

co
OM OAM &® HK OW Or An Ff & BH | O

co

‘oO

Oo
., Bn~T CaGnr & DN m

xCV

H*™,

Log pz
=A logl,

Log J,

4°822 906 5
4°813 6744
4°803 812 1
4°793 266 4
4°781 992 2
4°709 864 5
4°756 780 6
4°742 638 7
4°727 3298
4°710 7349
4°692 820 5
4°673 537 ©] 9979 289 3
4°652 826 3 | 9°977 7744
4°630 600 7 | 9'976 242 7
4°606 843 4|

4°581 198 5
4°553 312 5
4°522 705 0
4488 8749
4°451 310 4|
4+°409 781 0| 9°955 032 6
4°304 8136 9°951 1567
4°315.970 3| 9°947 097 4
4°203 067 7 | 9°942 894 1
4°205 961 8| 9°937 989 3
4°43 951 1 | 9932 1429
4076 O94 0 | 9°925 293 5
4001 387 5 | 9°918 370 3
3°919 757 81 g'g10 702 6
37830 460 4 | 9°903 699 1
3°734 159 5 | 9°897 6900
3°631 849 5 | 9°892 2869
3°524 136 4| 9°885 796 7
3°409 933 1! 9881 213 7
3°291 146 8 | 9°873 2061
3°104 352 9 | 9857 662 8
3°022 015 7| 9" 837 1226
2°859 138 3: 'Q° 812 034 5
2°671 1728 9766 5778
2°437 750 6' 9692 583 2
2°130 333 8 9'55y 862 3
1'6G0 196 1 | 9°264 046 4
0°9542425| —o

aw 3

9°99° 137 7
9°989 454 3
9°988 725 8
9°987 872 3
9°986 9161
9°985 8581
9°984 691 1
9°983 405 1
9°982 085 6
9°980 716 5

9°990 7679

id
¢
’
Ne ee aa ame ed

Healthy Inves —Male.— Adjusted Table,

Prob. Sur-

viving 1 Year.

Px ,
"978 966 5
"977 5471
"9760102
"974 3742
"972 461 3
"970 322 4
"967 961 6
"965 363 8
"962 509 8

"959 5897
"956 569 4
"953 4312
"950 111 2
"946 7659
942 660 0
937 808 2
"931 949 8
"925 0600
"917 1398
"908 804 7
"901 6387
"893 628 1
885 3140
876 787 «
"866 O40 5
855 3481
‘B41 963 9
"828 648 3
814.1465
"801 1229
"799 T14 3
"780 345 5 |
768 7706
"760700 4
*746 803 1
"129 5479
687 262 3
648 686 0:
"584.2217

"492 700 BY
"362 9629 |
‘183 673 5

*000 000 O

Prob. Dying |

in 1 Year. |

Vr

021 033 5
"022 4529
"023 989 8
"025 625 8
027 538 7|
"029 677 6!
"032 038 4'
"034 6362
"037 490 2,
"040 410 3'
"043 430 6,
"046 568 8
"049 888 8
053 234.1
"057 340 0:
062 191 8
068 o50 2!
"074. 940 0;
‘082 860 2:
O9T 195 3
‘098 361 3]
"106 3719)
"114 6860,
"123 212 9
"133 059 5;
"144 651 QI
158 036 1'
"171 3517)

1858535

"198 877 |
209 885 7 i
"219 654 5:
"2312294
"239 299 6
253 1969
"279 4521
"312 73771
381 3140)
"415778 3,

"597 299 2

"037 037 1
816 326 5
17000 000 o

a

XCvi

HF.
Healthy Inves.—Female.—<Adjusted Table. °

Prob. Sur- | Prob. Dying

>. oOo G9 i] (+)
- © DI An Rw WH ONO OXF ANP WHOM) DWYIOANADO HMO

hw bP HOO Or Anh BHO

wr

65 188
64213
63 210
62 173

a a

1003
1037
108!

4°998 634 2
4°996 800 5
4°994 563 5
4°992 000 9
4°989 1961
4°986 216 2
4°983 143 5
4°980 062 5
4°977 0556
4°974 138 5
4°97 1 141 3
4°967 964 3
4°964 509 5
4°960 6610
4°956 305 4
4°951 043 5
4°946 673 3
4°O41 SIT 4
4°936 262 3
4°931 0407
4°925 863 6
4°920 749 3
4°915 695 2
4°910 603 I

6| 4°905 4829

4°900 345 3
4°895 1462
4°889 884 1
4°884 557 5
4°879 1647
4873 680 9
4°868 ro09 3
4°862 453 4
4°856 704.7
4°850 854 5
4°844 899 5
4°838 880 6
4°832 809 0
4°826 683 7
4'820 490 5

5] 4°814 167 7

4°807 623 0
4°800 785 8
4°793 601 8

9°998 634 2
9°998 166 3
9°997 763 0
9°997 4374
9°997 195 2
9°997 0201

9°995 7944
9°995 278 1
9°995 029 8
9°994 8381
9°994 7599
9°994 7784
9°994 822 9
9°994 885 7
99949459
9°994 907 9
9°994 879 8
9'994 862 4
9°994 800 9
9°994 7379
9°994 973 4
9°994 607 2
9°994 5162
9°994 428 4
9°994 3441
9°994 251 3
9°994 149 8
9°994 045 0
9°993 981 1
9°993 928 4
9'993 8747
9°993 8068
9'°993 577 2
9°993 455 3
9°993 1628
9°992 8160

9°992 382 5

"996 8600
"995 786 8
"994 862 3
"994.116 8
"993 562 5
"993 1620
"992 949 8
"992 93°99
"993 100 4°
"993 395 5|°
"993 1224
"992 7114)"
"992 076 6]°
"99117771"
"999 1577)"
"989 186 31°
988 6210
"988 1847] °
"987 986 3
"988 048 6|°
988 1502
988 292 8|-
"988 430 2
988 343 3
988 279 5|°
"988 2400
988 1000
"987 9567 |°
"987 809 9
"987 959 5
"987 452 5
"987 2529
"987 061 2
"986 8504
"986 619 6°
986 3817]:
986 2366|°
°986 1169]°
"985 995 0
"985 840 8 |°
"985 5407
"985 043 3
984 380 1
983 5944 |"
"982 6130

Pz

eee

"014.005 0

viving 1 Year.| in | Year. .

|
qe |
|

"003 £400!
"004 213 2
"005 1377
"005 883 2
"006 437 5
"006 838 0
"007 050 2
"007 069 I

006 899 6
006 694 5

7006 877 6!

007 288 6
007 923 4
008 8223
009 842 3
0108137)

"O11 3790!

o51 815 3|

"012 013 7,

OST OSI 4’

‘o11 849 8

O11 707 2

"OIL 569 8
"O11 6567

O11 720 5

‘O11 7600
‘O11. 9000)

O12 043 3 |
O12 1901
O12 340 5
O12 547 5
0127471
0129388)
013 149 6
013 380 4,
013 618 3
013 763 4|
013 883 1 |

O14 159 2!

O14 453 3
"014.9567
"O15 6199

016 4056

‘017 3870}

|

xcvii

HF.
Healthy [nves.—Female.— Adjusted Table.

N Decre- Prob. Sur- | Prob. Dyi
Age. Living. ment Log I, Log pz viving 1 Year| in] Year.
F L, d. =A log /, Pz Qn
55 {61 092) 1116} 4°785 984 3 | 9°991 993 2 | "981 732.5] °018 267 5
6 159976) 1144) 4°777 977 5 | 9991 636 x | 980925 7| *019 074 3
7 | 58 832| 1170| 4°769 613 6! Q'991 2761 |"Q80 1129] *o19 8871
8 | 57 662) 1196 | 4°760 889 7 | 9°990 897 3 |'979 2584] -020 7416
9 | 56 466) 1331 | 4°751 787 0| 9990 427 4 | "978 199 3| *021 8007
60 | 55 235| 1308 | 4°742 214 4| 9°989 591 9 | °976 319 4| 023 6806
I | 53927 | 1395 |4°731 806 3 | 9°988 617 6 |°974.131 7] *025 868 3
2 | 52 532] 1495 | 4°720 423 9| 9'987 461 2 | "971 5412] 028 4588
3 | 51 037) 1601 | 4°707 885 1 | 9°986 158 2 | "968 6306] -031 36904
4 | 49 436| 1706 | 4°694 043 3 | 9°984 741 8 | 965 4907} 034 509 3
65 |47 739] 1784 | 4°678 791 4 | 9°983 456 3 |"9626231| °037 3769
6 | 45 946| 1846 | 4°662 247 7 | 9°982 1909 | "959 822 4) °040177 6
7 |44 100] 1914 | 4°644 438 6 | 9°980 729 7 |°956 5986) °043 401 4
8 | 42 186] 1982 | 4°625 168 3 | 9°979 101 0 | "953 0176] *046982 4
Q | 40 204] 2050 | 4°604 269 3 | 9°977 2708 | "949 0100] *o50Q900
70 138. 154| 2123 |4°581 540 1 | 9°975 1362 | "944357 1| °055 6429
I | 36.031 | 2232 | 4°556 676 3 | 9°972 227 6 | "938.053 3] °0619467
2 |33 799 | 2338 | 4°528 903 9 | 9°968 868 6 | 9308264! ‘069 173 6
3 | 31 461 | 2425 | 4°497 772 5| 9°905 164 3 |"9229204| °077079 6
4 | 29.036 | 2490 | 4°462 9368 | 9°961 062 3 | 914.2444) 085 755 6
75, | 26 546 | 2518) 47423 9091/1 9°956 718 5 | "905 1458} “0948542
6 |24 028 | 2500] 4°380 717 6] 9°952 2861 |°8959547| “104045 3
7 |21 528 | 2363 | 4°333 003 7 | 9°949 505 I |°890 2360) ‘109 7640
8 | 19 165 | 2205 | 4°282 508 8} 9°946 917 0 | 884.946 5] "115053 5
9 | 16960] 2024 | 4°229 425 8| 9°944 808 5 | 880 6604) “119 339 6
80 | 14.936] 1819 | 4°174 234 3 | 9°943 600 2 |°878 213 7| 121 7863

13 117| 1621 | 4°117 834 5 | 9°942 712 3 |°8764199| °123 5801
11 496| 1514] 4060 546 8 | 9°938 670 8 | 868 302 0| +131 6980
9 982 | 1450! 3°999 217 6| 9°931 833 2 | "854.738 5| "145261 5
8 532} 1389 | 3°931 0508 | 9°922 829 9 | 8372011] "1627989
7 143] 1326 | 3°853 8807 | 9°910818 4 | 814 363 8! +185 6362
1234 | 3°764 699 1 | 9°896 450 8 | -787 863 2| °212 1368
4 583 | 1086 | 3°661 1499 | 9°882 545 7 | °763 037 3) °236962 7
3.497| 903 | 3°543 695 6| 9°8702744 |°741. 7786) -2582214
2594] 707 |3°413 9700] 9°861 Bol g | "727 448 0) °272 5520)
1887] §519|3°275 7719 | 9860 314 2 |°724. 9603] °2750397

CO
0'O C™ Qurr bh GD B mm O
tn
Cc
bone
~

ve)

1 | 1368) 368] 3°136 0861 | 9°863 913 9 |°7309942] °269005 8
2 | 1000| 2321] 3°000 0000; 9'885 361 2 |°768 0000! ‘232 0000
3} 768) 138} 2°885 361 2| 9'913 979 3 | 820 312 5| *179 687 5
4 | 630] 11712°799 340 5| 9°910 776 9 |°814 285 7| °186 714 3
95 513] 107/2°710 117 4| 9°898 408 6 |°791 423 0] *208 5770
6 406| 104] 2°608 5260| 9°871 480 9 |°743 842 4| °2561576
7 302] 102] 2°480 0069| 9°821 023 1 |'6622516| 3377484!
8 200] 100] 2°301 0300] 9°6 989700 |*5000000| .5000000
9 100} 100] 2°000 0000 —o "000 000 0} I'000 000 0 |
100

Oo — |

xeviii
H™,
Healthy Lives.—Male and Female.— Adjusted Table.

No.
Living.
le

Decre-
ment,

4,

Log ls

Log Ps
=Alog ly

Prob. Sur- | 1
lriving | Year,
Ps

- w & » » “ 6
Rew So wr ah ae on Bo Or hae wn So Or anne wn SO OV Ana nnd

100 000
99558
99451
98 766
98 390
g8orr
97 615
97 189
96720
96 195
95 614
94993
94.348
93 695
93 044
92 397
91750
91099
90.431
89 745
89 042
88 324.
87 598
86 865
86 122

85 368;

84.600
83 811
83.000
82170
81 326
80472
79 612
78 743
77855
76942

5°000 000 0
4°998 0762
4996297 1
+994 607 5
49929510
49912748
4989 5166
4°987 6171
4985 5163
4°983 152 5
47980 521 5
4977 6916
49747327
49717164
4°968 688 4
4965 9579
4°962 606 1
+959 513 6
4°956 3173
4953 010 3
4949 5949
49460787
4°942 4942
47938 844 8
4935 1141
49312951
4927 3704
4°923 3010
49190781
49147133
49102294
4°905 6448
4°900 978 5
4/896 2120
4°891 286 5
4°886 163 5
4880 779 3
4°875 0902
4869 090 8
4/862 781 1
4856 166 8
4849 2597
4°842 028 5
4834 439 8
4826 4117

9°998 0762
9°998 2209
9°998 3104
9°998 343 5
97998 3238
97998 2418
9°998 100 5
9°997 899 2
9°997 6362
9°997 3690
9°997 1701
9°997 0411
9°996 983 7
9°996 9720
9°996 969 5
9996 9482
9°996 907 5
9°996 803 7
9°996 693 0
9°996 584.6
9°996 483 8
97996 415 5
97996 3506
97996 269 3
9°996 1810
9°996 075 3
9'995 9306
99957774
9°995 6352
9995 5164
99954154
9°995 3337
9°995 233 5
9°995 0745
9°994 8770
9°994 6158
9°994 3109
9°994 000 6
9°993 690 3
9°993 3857
9°993 0929
9°992 768 8
9°992 4153
9'991 9719
9°991 4708

"995 580 0| "004 4200
"995 911 9| 004 088 1
"996 117 0] 003 883 0
"996 193 0| 003 807 0
"996 148 0] 003 8520
"995.959 6| 004 040.4
“995 635 9] "004 364 1
"995 174 4| "004 825 6
“994 572 0] :005 4280
"993 960 2] 006 039 8
"993 505 1]°006 4949
"993 210 0| 006 7900
"993 078 8|-006 921 2
"993 051.9] ‘006948 t
"993 046 3 |°006953 7
"992.997 ©| 007 002 4
"992.904 6] *007 095 4
"992 667 3 | 007 3327
"992.414 1|'007 585 9
"992 166 7 | "007 833 3
"991 936 4|*008 063 6
"991 780 3 |°008 219 7
"991 632 2| "008 367 8
“991 446 5|"v08 553 5
"991 245 0| 7008 7550
"991 003 7 |"008 996 3
"990 673 8| ‘009 3262
"990 323 5|*009 676 5
*990 000 00100000
"989 7286 ‘oro 271 4
"989 499 1 | 010 5009
+989 313 1|"0106869
+989 0846 |‘o10915 4
+988 722 8)o1t 2772
+988 273 1|"011 7269
"987 679 0|*012 3210
"986 985 8] 013 0142
*986 2809] °013 7191
"985 576.4 |°014 4236
"984 885 3] 015 1147
+984 221 6) 015 7784
"983 487 6|°016 5124
"982 678 0} ‘017 3220
+981 684 6] ‘018 315 4
“980 552 4|°019 4475

in
in

Spo wn OO GAT QO

~y

\O Ve) foe) co ~J
OO On An P ® BH et OLO CONT Qin & © DH HOM} CNT Ant © OH ONO COX OA

o
ie)

Age Living. | ment.

No. _ | Decre-

1358
1414
1471
1531
160r
1677
1760
1849
1936
2014
2080
2138
2186
2224
2268
2331
2401
2469
2531
2567

33

XC1X

Hr

Log /,

4°817 882 5

4°808 818 4
4°799 175 1
4°788 910 4
4°777 963 0
4°766 2120
4°753 552 4
4°739 8570
4724 988 2
4°708 854 1
4°691 408 6
4°672 624 2
4°052 4301
4°630 763 4
4°607 551 5
4°582 529 2
4°555 215 4
4.525 161 5
4°491 9217
4°454 982 0
4°414 003 5
4°369 2159
4°320644 7
4°268 461 0
4'212 4007
4°152 227 2
4°087 3909
4°017 283 8
3°941 2132
3°858 055 7
3°797 304 3
3°668 944 7
3°562 5308
3°447 023 1,
3°325 1050
3°194 7918
3°053 462 6
2°900 367 I
2°738 780 6
2°564 666 1
2°372 9120
2°176 og! 3
1°973 1279
1°698 9700

17| 1°230 4489

au= OO

9°990 9359 |"
9°990 3567 |°
9°989 7353 1°
9°989 052 6 |"
9°988 249 0 |°
9°987 3404 1°
9°986 3046 |°
9°985 1312 ]°
9°983 865 9 |-
9°982 554 5 |"
9°981 2156 ]°
9°979 805 9 |°
9°978 333 3]°
9°976 788 1 |°
9°9749777|°
9°972 686 2 |°
9°969 9461 |°
9°966 7602 |:
9°963 060 3 | °
9°959 021 5 |°
9°955 212 4 |°
9°951 4288 |:
9°947 816 3 |°
9°943 9397 |°
9°939 826 5 |°
9°935 1637 |°
9°929 8929 |°
9°923 929 4 |°
9°916 842 5 |°
9°909 2486 |-

9°901 6404
9°893 586 1
9°885 092 3
9°877 4819
9°869 686 8
9°858 6708
9°846 904 5
9°838 413 5
9°825 885 5
9°808 2459
9°803 179 3
9°797 0366
9°725 842 1
9°531 4789

md °°)

Prob. Sur-

Healthy Lrves,—Male and Female.— Adjusted Table.

Prob. Dying

viving 1 Year| in 1 Year.

"797 3342
*782 683 2
"797 5247
“7541919
‘7407758
"722 2222
"792 9177
689 308 1
"669 708 o
643.051 8
635 3932
"626 666 7
"5319149
*340 000 0
"000 000 0

fz

0206546

"027 9599
"023 3581
"024 892 3
"026 694 9
‘028 7290
"031 042 §
"033 657 3
"036 468 4
"039 373 6
"042 3306
"045 434 3
"048 665 4
"052 O44 1
eas
060 9
"066 861 7
°073 681 7
"O81 5400
"090 O41 7
"097 987 8
"105 8119
"113.219 3
"121 099 4
"129 384 4
"138 681 5
"149 0719
"160 6766
"174261 5
"188 5746
"202 665 8
"217 3168
"232 475 3
"245 808 x
"259 2242
"2777778
"297 082 3
"310 6919
°330 2920
"356.948 2
"364 406 8
"373 333 3
*468 085 1
*660 000 0
T°000 000 0

TABLES.

ONE LIFE.

—EEe ee
CO —————— ———

H™,

HEALTHY MALE LIVES.

Lan ime

we

wo tS)
WO WOr~3 An POH HO OHO WT AN PW! vw WO /}O W~ 7 Aw Pw WD HO

9

Sas >> >
0 OO On~T AM RO DY | O

G Db

2
H™

Elementary Values.

Log t,

5°000 000

4°997 867

"996 131
"994 687
"993 419
"992 218
"990 969
"989 557
"987 867
"985 781
"983 279
"980 521
‘977 595
"974 608
971 661

"968 768
"965 879
"962 966
"959 957
"956 831

953 591
"950 224
"946 771
"943 237
"939 624

"935 915
"932 088
"928 14
"923 984
"91g 716
"915 315
"910 816
"906 238
"QOL 551
"896 692
"891 643
886 316
"880 659
"874 667
868 350
"861 690
"854 707
"847 406
839 717
"831 563

1997 867
"998 264
"998 556
"998 732
"998 799

"998 751
‘998 588
"998 310
"997 914
"997 498

997 242
"997 070
"997 017
"997 053
"997 107

"997 {II
"997 087
"996 991
"996 874
°996 760

"996 633
"996 547
"996 466
"996 387
"996 291
"996 173
"996 026
"995 870
"995 732
"995 599

"995 501
"995 422
"995 313
"995 T41
_ 994951

"994 673
"094 343
"994 008
"993 683
"993 340°

"993 O17
"992 699
"992 311
"991 846
"991 344

000 397
000 292

000 176
000 067

999 952

999 837
999 722
999 604
999 584
999 744

999 828
999 947

000 036
000 054
000 004

999 976
999 904
999 883
999 886
999 873

999 914
999 919
999 921
999 904
999 882

999 853
999 844
999 862
999 867
999 902

999 921
999 891
999 828
999 810
999 722

999 670
999 665
999 675
999 657
999 677

999 682
999 612
999 535
999 498
999 423

Colog #,

5°000 000
002 133
"003 869
"005 313
"006 581

"007 782

"009 031

"O10 443
"O12 133
"014.219
7016 721
"019 479
"022 409
"025 392
028 339
"031 232
"034 121
"037 934
"040 043
"043 169

‘046 409
"049 776
"053 229
"056 763
"060 376

"064 085
"067 O12
"071 886
"076 016
"080 284

"084. 685
"089 184
093 762
"098 449
"103 308

"108 357
"113 684
"1IQ 341
"125 333
"131 650

"138 310

"145 293

"152 594

"160 283
"168 437

Zz

a an

ww

wo w
WO COr~ 7 Qn PW OW HO WO DWr~ At HON HO LO Cr Qn PO WB HO

©o

bwenad

tin
hObYH HO HO CON AN

0°002 133

on pow nS

in
POH mM O OO DOr ANH

"O01 736
"OOl 444
"001 268
"OOl 201

"OO! 249
"OO! 412
"001 690
"002 086
"002 502

"002 758
"002 930
"002 983
"002 947
"002 893

“002 889
"002 O13
"003 009
"003 126
"003.240

"003 367
"003 453
"003 534
"003 613
"003 709
003 827
003 974
"004 130
"004 268
"O04 401

"004 499
‘004 578
"004 687
"004 859
"005 049

"005 327
"005 657
"005 9g2
"006 317
"006 660

006 983
"007 301
"007 689
"008 154
"008 656

Elementary Values.

999 933
000 048

000 163
000 278
000 396
000 416
000 256

000 172
000 053
999 964
999 946
999 996

000 024
000 096
000 II7
000 114
000 127

000 086
000 081
000 079
000 096
000 118

000 147
000 156
000 138
000 133
000 098

000 079
000 109
000 172
000 190
000 278

000 330
000 335
O00 325

000 3.43
000 323
000 318
000 388
000 465
000 502

000 577

H™,

397
329
288
272

282
318
379
466
556
609
643
650
638
622

617
618
634
654
673
694.
706
717
727
740
757
779

802
821
838
848
854
865
887
gil

95°
996
I O41
xr 082
1 124

1 160

I 193
I 235
1 286

I 339

Log 4,

490|,2°690 196

"598 791
"517 196
"459 392
"434 569

"450 249
502 427
"578 639
*668 386
"745 975
"784 617
“808 211
"812 913
"804 821
"793 79°

"790 285
“7y0 988
"802 089
"815 578
"828 015

"841 359
"848 805
"855 519
"86 534
"869 232
879 096
"B91 537
"904 174
"O14 343
"923 24y

"928 396
"931 458
"937 016
"947 924
"959 518

"O77 724
"998 259
3°017 451
"034227
"050 766

"064 4.58
"076 640
"og! 667
"109 241
"126 781

A

908 595
918 405
942 196
975177
O15 680

052178
076 212
089 747
076 689
039 542

023 594
004 702
991 908
988 969
996 495
000 703
OII 10]
013 489
O12 437
O13 344

007 446
006 714
006 O15
007 698
009 864

O12 441
O12 637
O10 169
008 gol
005 152

003 062

005 558

O10 go8

OIT 594
018 206

020 535
O19 192
016 776
016 539
013 692
o12 182
O15 027
017 $74
017 540
O19 037

O9I 405
081 595
057 804
024 823
984 320

947 822
923 788
O10 253
923 311
960 458

976 406
995 298
008 092
O11 031

003 505

999 297
988 899
986 511
987 563
986 656

992 554
993 286
993 985
992 302
ggo 136

987 559
987 363
989 831
99T 999
994 848

996 938
994 442
g89 092
988 406
981 794
979 465
980 808
983 224
983 461
986 308
987 818
984 973
982 426
982 460
980 963

bh |

we

©» ©)
© cer An POH HO OO ONAN PO HHO LO Or Qn HO DW =» O

aX)

aN
&& DH O

>

—& OO Be O WO COM AN

=

FS pw wn

ee] fo ¢) ~J ~Y
WO CO~NrAMH HW He GO LW GW TAN HON MO WO OH” A

& & DY

‘Oo
“IAN

66 513
O65 114
63 652
62 125
60 533

58 866
57 119
55 289
53 374
51 373

49 297
47 156
44 960
42717
40 443

38 124
35 753
33 320
30 823
28 269

25 691
23 164
20 700
18 326
16 068

13 930
IIQIS
10 032
8 313
6 768

5 422
4 284
3 343
2 57°
1955
1 460

Log f;

4°822 907

"813 674
"803 812
"793 266
"785 992
~769 865
"756 781
"742 639
"727 33°
710 735
692 820
‘673 537
652 826
"630 Gor
606 843
581 198
"553 312
"522 705
"488 875
"451 310
"409 781
"304 814
"315 970°
"263 068
"205 962

"143 951

"076 094
"001 388

3°919 758

"830 460

"734 160
631 849
524 136
"499 933
"2O1 147
"164 353
"022 016

2°859 138

"671173
437751
"130 334

1°690 196
0°954 243

4
HM,

Log Pr

1990 767
"990 138
"989 454
"988 726
"987 873
986 916
"985 858
984 6g!
"983 405
982 085

‘980 717
"979 289
°977 775
"976 242
"974 355

"972 114
‘969 393
°966 170
"962 435
"958 471

"955 933
"951 156
"947 098
"942 894
"937 989

"932 143
"925 2904
°918 370
"910 702
"903 700
"897 689
"892 287
"885 797
"881 214
873 206
"857 663
837 122
"812 035
"766 578
"692 583
"559 862
264 047

Elementary Values—(continued.)

A

999 371

999 316
999 272
999 147
999 943

998 942
998 833
998 714
998 680
998 632

998 572
998 486
998 467
998 113
997 759

997 279
996 777
996 265
996 036
996 562

996 123
995 942
995 796
995 995
994 154
993 151
993 076
992 332

‘992 998

993 989

994 598
993 510
995 417
991 992
984 457
979 459
974913

954 543
926 005

867 279
704 185

Colog 2,

5°77 993
"186 326
"196 188
"206 734
°218 008

"230135
"243 219
"257 361
*272 670
289 265
*307 180
"326 463
"347 174
"309 399
"393 157
"418 802
*446 688
"477 295
"511 125
"548 690
"590 219
635 186
"684 030
"736 932
"794 038

"856 049
"923 906
998 612
4°080 242
"169 540
"265 840
"368 151
"475 864
*590 067
708 853

835 647
_"977 984
3°140 862

328 827

"562 249

"869 666
2°309 804
1045 757

qn
WO co~nr Qin

A A
© WH O

“wy
OO cor~7 Cin PO HD O LW Or~ 3 Quan

Ve) 0o (o2) ~}
OO CON AM PW WH HO

Pwpr oO

\O
“IAN

in
- Fre vn8 oor ot | &

~r

6
|
8
9
°
I
2
3
4
5
6
7
8
9

co oe]
Bo wow akd P&H 1 O

wae f O DN HH

Colog pz

0°009 233
"009 862
"O10 546
"O11 274
"O12 127

"013 084
"O14 142
"O15 399
"016 595
017 O15
"019 283
"020 7II
"022 225
"023 758
"02.5 645
"027 886
"030 607
"033 830
037 565
O41 529
"044 967
"048 844
"052 902
"057 106
-°062 O11
"067 857
"074 706
"081 630
089 298
"096 300
"102 31r
"107 713
"114 203
"118 786
"126 794
"142 337
"162 878
"187 965
"233 422
"397 417
"440 138
735 953

5
H™

Elementary Values—(continued.)

A

000 629
000 684
000 728
000 853

000 957

oor 058
oor 167
oor 286
OOL 320
oor 368

oor 428
OO! 514
OO! 533
oo1 887
002 241

002 721
003 223
003 735
003 964
003 438
003 877
004 058
004 204
004 905
005 846
006 849
006 924
007 668
007 002
006 or!

005 402
006 490
004 583
008 008
O15 543

020 541
025 087
045 457
073 995
132721

295 815

ad, | Logd,

I 399
I 462
I 527
I 592
1 667

1747
1 830
Tgt5
2 Oo:
2 076

"104 947
"183 839
"201 943
"221 936

"242 293
"262 451
282 169
"303 247
317 227

*330 617
"341 632
"350 829
"356 790
"305 501
"374 932
*386 142
"397 419
"407 221
"411 283

214!
2 196
2243
2274
2 319

2 371
2 433
2 407
2 554
2578
2 527
2 464
2 374
2258
2 138

"402 605
391 641
"375 481
353 724
"330 008

"304 275
274 850
"235 276
"188 928
"129 045
"056 142
2°973 590
888 179
788 875
"694 605
"610 660
"517 196
"404 834
"290 035
"143 O15
86 1°934 498
40! ‘602 060

9} 9°954 243

2015
1 883
1719
1 545
I 346
1 138
941
773
615
495
408
329
254
195
139

3°145 818

A

O19 129

018 892
018 104
O19 993
020 357

020158
org 718
019 078
o15 980
O13 399
O11 O15
009 197
005 96:
008 511
009 631

OII 210
O1l 277
009 802
004 062
Q9I 322

989 036
983 840
978 243
976 284
974 267
97° 575
960 426
953 652
940 117
927 097
917 448
914 589
goo 696
995 73°
916055
906 536
887 638
885 201
852 980
791 483
667 562
352 183

—A

980 871

981 108
981 896
g80 007
979 643
979 842
980 282
980 922
984 020
986 610

988 985
990 803
994 939
991 489
99° 369

988 790
988 723
ggo 198
995 938
008 678
O10 964
©16 160
O25 757
023 716
025 733

029 425
039 574
046 348
059 883
072 903
082 552
085 411
099 304
094 270
083 945
093 4064
112 362
114 799
147 020

208 517

332 438
647 817

in
OO Gs~7 An

WN
.@)

\O ioe) Cc ~T ~T! ~A
0 OO G7 Qn POH ew O WO Wr QAKr HW HD CO CO CGO~ AN ADO D &

Pw bP

Le)
~s CAG

Probabilities of Living over, and >

ration of

|

=

wo

Oo

Go

fe)
I
2
3
4
5
6
7
8
9
)
I
2
3
4
25
6
7
8
9
°
I
2
3
4
5
6
7
8
9

iw,
ro] © wx Oth pwn nd

~~ &W WY

Average

"995 100
"996 O10
"996 681
997 085
997 238

"997 129
"996 753
"996 118
"995 208
"994 255

"993 671
"993 275
"993 156
"993 236
"993 361

"993 370
"993 315
"993 096
992 828
"992 567

"992 277
"992 083
"991 895
"OOT 715
"991 496
"991 226
"990 8g!
"990 536
"990 220
"989 918

"989 694
"989 513
"989 266
"988 873
"988 444
"987 808
"987 060
"986 298
"985 560
"984 780

"984 050
"983 330
"982 451
°981 400
‘980 266

"002 871
"003 247
"003 882
"004 792
"005 745
"006 329
7006 725
"006 844
"006 764
"006 639

“006 630
"006 68 5
"006 go4
“007 172
"007 433

"007 723
"007 917
“008 105
"008 285
"008 504

"008 774
"009 10g
"009 464
"009 780
"010 082

"O10 306
"010 487
"O10 734
‘OII 127

“OII 556

"O12 192
"012 940
"O13 702
"014.440
"O15 220

"015.950
"016670

"O17 549
"018 600

"O19 734

Oe ws om

~r
&Oowvwm O OW CT A

~

co
© on Ar SPO Hm O YO ON Qn

1° 2)

‘Oo

he)
“saat PO WP HO

g

fe,

"004 YOO | 50°
"003 990
"003 319
"002 O15
"002 762

"979 322
"967 962
"995 364
"962 510
"959 59°

"956 569
"953 431
"Q5O 1
"946 766
"942 660

"937 808
"931 95°
"92,5 060
"917 140
908 805

"QOT 639
"893 628
"885 314
876 787
"866 O41

"855 348
"841 964
"828 648
"814 147
801 123

"790 II5
"780 345
°768 770
"760 700
"746 804
"720 548
687 263
648 686
"584 222
°492 700

"362 964
"183 673

ing in, a Year, and the
oe each A ge

Tz
(1—pz)

"021 033
"022 453
"023 9QO
"025 626
027 539

"029 678
"032 038
"034 636
"037 49°
"040 410

"043 431

"062 192
"068 050
"074940
"082 860
"091 195
"098 361
"106 372
"114 686
"123 213
"133 959
"144652
"158 036
"171 352
"185 853
198 877
"209 885
"219 655
"231230
"239 300
"253 196

°279 452
"312737
"351314
415778
"507 300
637 036
°816 327
"000 000 [5°000 000

H™,

THREE PER-CENT.

Lm! bout

wo

©o

&o we
WO Cn~T Qin BPW HD Mm CO CO DOr? AN HMO OHO WO ONAN HPO HD mm O

pwend

pa

Or
—&woObK eM CO OO Cr Qn

74 400'4
71 888'1
69 515°9
67 267°!
65 117°5
63 046°2
61 034'2
59 064°!
57 121°2

55 1907

53 276°3
54 3972
49 504°7
477907
46 085°8
44 446°5
42 865°8
41 339°1
39 857°9
38 419°5
37 02 3°2

35 667°3
34354°2
33 083°3
31 853°6

30 662°8
29 508°6
28 388°1
27 300°5
26 246°1

25 224°7
24 237°6
23 284'9
22 364°0
21 471°!

20 604°8
Ig 760°8
18 936°9
18 133°5
17 35Ul
16 589°3
15 849°2
15 131°!
14 4326
137516

8
H™.,

3 PER-CENT.
Commutation Table.

I 796 867
1724979
1655 403
I 588 196
I §23 079
I 460 032
I 398 998
T 339 934
¥ 282 813
1 227621

1174345
1 122947
1073 383
1025 59!

979 5§95°3
935 058'8
892 193°0
850 854'0
810996"!
772 576°6

735 553°4.
699 886°1
665 531°9
632 448°6
600 595°0

569 932°2
540 423°6
512 035°5
484 735°0
458 489°0
433 264°2
409 026°6
385 741°7
363 377°6
341 906°6
321 301°8
301 541°0
282 604°!
264 470°7
247 119°6
230 530°3
214 681'0
199 549°9
185 117°3
171 3658

36 413 646
34 616779
32 891 800
31 236 337
29 648 141
28 125 062
26 665 030
25 266 032
23 926 098
22 643 285
21 415 664
20 241 319
19 118 371
18 044 988
17 O19 397
16 039 892
15 104 833
14 212 640
13 361 786
12 550790
11778214
11 042 660
10 342 774

9 677 242

9 044 794

8 444 199

7 874 266

7 333 843

6 821 807

6 337072

5 878 583
5 445 319
5 036 292
4.650 551
4287 173

3945 267
3 623 965
3 322 424
3 039 820
2775 349
2 528 229
2 297 699
2 083 018
1 883 468
T 698 351

19 906°2

19 552°2
19 273°8
19 049°7
18 859°3
18 684°7
18 509.0
18 316°6
18 o94'0
17 828°2

17 520°4
17 193°0
16 857°4
16 528°1
16 214°2

159172
15 6311
15 352°9
15075'8
14 798°2
14 521°0
14 243°4
13, 969°2
13 698'9
13 432°8
13 169'8
12 908°6
12 647°6
12 386°8
12 127°6

11 870°7
II 618°3
II 371°5
rr £28°8
10 887°3

10 646°3

10 402°4

10 154°2
9 902°27
9 64805
9 391°66
9 134°76
8 878:25
8 620°44
8 359°81

756 181°7

736 275°5
716 723°3
697 449°6
678 399°8
659 54°°5
640 855°8
622 346°7
604 030°1
585936":
568 107°9
55° 587°5
533 394°5
516 537°0
500 008°9

483 794'7
467 877°5
452 246'5
436 8936
421 817°9

407 0196
392 498°7
378 255°3
364 286°
350 587°2

337 154°5
523 984°7
311 0761
298 428°4
286 0o41°6

273 914°1
262 043°4
250 425°1
239 053°6
227 924°7
217 037°5
206 391°!
195 988°7
185 834°5
175 932'2
166 284'2
156 892°5
147 757°8
138 879°5
130259'1

a. o> & ee S = =
0 OO ONAN PO YHOO HO CONAN HW BK EO LO Gr? An Pw WY =O

OK pwen

iw
Hove O OO CONTA

CO —— anni <

z D,

55|13 087°6
6|12 439°!
7111 805°7
8i1x 186°8
g|10 582°7
999F51
9 412°61
8 845°67
8 290°57
7 147°34

7 217°74
6 703°17
6 204°86
5 723°60
5 261°08

4 814°96
4 383°99
3 966°66
3 562°52
3.172°17
279891

2450°10

DP pow D

~

“I

9
HM™,

3 PER-CENT.
Commutation Table—(continued),.

Nz

158 278-2

145 839°!
134933°4
122 846°6
112 263°9

102 272°4
92 859°77
84 014'10
75 723°52
67 976°19
60 758°45
54055728
47 850°42
42 126°82
36 865°74
32050°78
27 666°79
23 700°Y3
20 137°61
16 965°44
14 166°53
II 716°43

Sz

I 526.985
1 368 707
1 222 868
1 088 834

965 987°7

853 723°8
751 451°4
658 591°6
574 577°5
498 8540

430 877°8
37° T19*4
316 064°1
268 213°7
226 086'9

18Q 221°%
157 170°3
129 503°5
105 803-4

M;

8 096°34
7 829°08
7 557°92
7 282°95
7 004°63
6 721°69
6 433°80
6 141°02
5 843° 56
5 541°80

5 237°85
4933°5°
4 630°44
4 329°90
4 034°08
374120
3 45°°47
3 160°83
2 872°23

85 665°80|2 585°63
68 700°36|2 304°77
54 533°83)2 037°49

|121 899°2

OO COr~7 Ain WH WO WO Dr T1 OA

OO
OO Cr? Qn huwpne

ve)
°

\©O
“ran # © b

2125°71
1827°11
I 555°32
I 309°10
1 087°12
888°659
714°938
_ 565111

439°537
337°170
255°445
190°659
140°810

102'095

714214
47°655 6
30°013 I
17°023 6
8°143 2
2869 6
“517

9 590°717

7 763°609)| .

6 208°285

4 899°184
3 812°061
2 923°402
2 208°464
I 643°3 5¢
I 203°817
866°647
611°202
420°543
279°733
177°638
106°217
58°561
28°548
11°525
3°381
512

“000

42 817-40|1 784°46
33 220°68)1 547°77
25 463°07|I 329°20
19 254°79|1 128°28

14 355°61
10 543° 54
7 620°14
5 411°68
3 768°32
2 504°51
1 697°86
1 086°66
666'11
386°38
208°74

944°429
777-628
629°790
500°787
391°672
302°107
2.30°203
172°857
128°561

93°947 0
66°247 5
44°5619
2B°307 4
16°192 I
7°807 6
2.9711
"4968

R, . |2

wn
© con~r Aw

113 802°9

105 973°8
98 41591
OT 132°95
84 128°32
77 406°64
7° 972°84
64 831°82
58 988°26

53 440°26
48 208°62
4327511
38 644°67
34 314°77
30 280°69
26 539°49
23 089'01
19 928°18
17 055°96

14 470°32
12 165°55
10 128°06
8 343°609] 8
6 795°842} 9
5 466°642/ 80
4 338°365) 1
3 393°937| 2
2616°3009| 3
1986°518] 4
1 485°732|85
1.094'060|] 6

791°953| 7
561°750] 8

388°893) 9
260°331}g0
166°384] 1
I00°I37} 2
55°575} 3
27'268) 4
11'076195
3°268] 6

497} 7

Cc

~r
OS pwn

wy
WIA FW RHO OO DOXA

10

Ho®™,
3 PER-CENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

A A
(Log vp;)| 1° P= \ Cologep,

te ee Pe eel

4°871 628/1°985 029 5°128 372/0°014 97116'254 5161982 268

Ln]

"856 657

we ww »p Sy me
BS ema &ObwN me O WO CH~3 An PHD = O [OO DW~ 1 Qn HON - O

PO bm

aN

Cnr
hOKH HM CO WO DOr AN

985 427
"842 084! -
"827 803] °
813 698] -

‘799 659) °
"785 573] °
"771 324) °
756 797| °
“741 874) °

"726 534] °
°{10 940 °
"695 172 °
"679 352) °
"663 567] °
"647 837) °
"632 111] °
"616 361) °
"600 514! *
"584 5551 °
"508 474) °
552 270] °
"535 980} °
"519 60g} °
"503 159] °
"486 612) °
"469 948) -
"453 137| °
"436 170) °
"419 064) °
"$01 826] -
"384 490] °
"367 075) °
"349 559 *
"331 854! °

"313 968) -
295 804] °
"277 310) °
"258 481) °
"239 327] *
"219 829] °
"200 008] °
"179 870] °
"159 3441 *
"138 3.53] °

"143 343! °
"157 916] °
"172 197] °
"186 302] °
"200 341] *
"214. 427) °
"228 676) °
"243 203! °
"258 126] °

"273 406) °
"289 060] °
"304 828) -
"320 648
"336 433

"352 163] °
"307 889) °
"383 639] °
"399 486] -
"415 449)

"431 526] °
"447 739 °
"464 020! °
"480 391) °
"496 841) °

"513 388) °
"530052) °
"546 863} °
"563 830] °
*§80 936] °
"598 174] °
“615 S10} °
°632.925| °
650.450] °
668 146] °
686 032] °
"704 196} °
"722 690} °
“741 519] °
700 673} -
"780 171| °
"799 992 °
"820 130] °
"840 656] °
"B41 647| °

"O15 785

236 7841982 136
218 920/981 984
"200 904/981 818
"182 722/981 641

"164 3631981 454
"145 817/981 266
"127 083/981 080
"108 1631980 gor
7089 0641980 732

"069 7961980 564
"050 360/980 395
030 7551980 219
"010 974/980 033
"O15 73015991 0071979 832

"97° 8391979 620
"95° 4591979 396
"929 8551979 164
"909 019/978 923
"887 9421978 672
866 6141978 413
"8450271978 142
"823 1691977 856
"B01 0251977 5.57
778 5821977 241

"755 823 976911
"732 7341976 566
"709 300/976 204
"685 5041975 825
"O61 3291975 424

636 7531974 999
"611 7521974 545
586 2971074 O61
"560 3581973 549
"533 997/973 006

"506 9131072 433
"479 3461971 833
"451 1791971 199
"422 3781970 529
"392 907 969 821
"362 7281969 066
"331 7941908 258
"300 0521907 395
‘267 4471906 477
"233 924|965 497

&> wo iS) N bat a
m Boma POH HS CODON POH NO OCOVON FH KHO

-Hbwb ee OC DO COn~I Qn PW HD

*

10 (7°745 484
"763 216
-781 080
"799 096
"817 278

"835 637
"854 183
"872 917
"891 837
"910 936

"930 204
"949 640
"909 245
| "989 026
6°008 993

029 161
049 541
070 145
"090 981
"112058

"133 386
"154973
"176 831
"198 975
-221 418

"244.177
"267 266
"290 700
"314 496
"338 671

"303 247
388 248
"413 703
"439 042
"406 093

"493 087
"520 654
548 821
°577 622
"607 093
"637 272
"668 206
"699 948

"732 553
*766 076

>. . wo & SY ) mm
O OAM” BH 8 HO OH ONAN PW HHO HO CON QA BW HHO W/O DONT AN BPW WS

in
Oo

-~ OW DH

Colog Nz

A

O17 732 |4°298 988 | 992 208 [5-701 012
"291 196
‘284 967
"279 889
275 526
"271 487
267 383
262 845
"257 534

"251 108

"243 544
"235 352

017 864
018 016
018 182
018 359
018 546
018 734
018 920

O19 099
019 268

O19 436
org 605
019 781
019 967
020 168
020 380
020 604
020 836
021077
O21 328

o21 587
021 858
022 144
022 443
022 759
023 089
023 434
023 796
024.175
024 570
025 OOI
025 455
025 939
026 451
026 994
027 567
028 167
028 801
029 47!
030179

030 934
O31 742
032 605
033 523
034 503

11
H™,

8 PERCENT.
Logarithms and Co-logaritims of D,, Nz, and M,; with

their Differences.

Log M,

226 7092

218 223
“209 897
"201 866
"193 989
°186 189

"178 279
"170 210

161 995
"153 613
"145 172
"136 685
"128 165

"119 578
"110 879
"102 009
"092 959
"083 774
"074.476
"065 142
7055 818
"046 450
036.918
"027 201
"017 136
006 645
3°995 735

"984 440

"972 742
"960 697
948 327
"935 53°
"922 196

A

993 771
994 922
995 937
995 961

995 896
995 462
994 689
993 574
992 436
ggt 808
991 440
991 431
991 674
991 969
992 123
992 200
992 090
991931
991 785
991 618
991 559
O91 513
991 480
99T 413

991 301
gg1 130
999 950
ggo 815
ggo 702
990 666
990 676
990 632
990 468
990 283
989 935
989 509
989 ogo

Colog M,

"708 804
"715 933
"720 11
"724 474

‘728 513
°732 617
“737155
"742 406
"748 892
"758 456
"764 648
"773 208
781777
"790 103

"798 134
"806 o11
"813 811
821721
"829 790
"838 005
"846 387
"854 828
"863 315
871 835
"880 422
"889 121
‘897 991
"907 O41
"916 226

"925 524
"934 858
"944 182
"953 55°
"963 082

"972 799
982 864
993 355

988 705 |4°004 265

988 302
987 955
987 630
987 203
986 666
986 093

"O15 560

027 258
039 303
"051673
"064.470
"077 804

A

007 792

006 229
005078
004 363
004 939
004 104
004 538
005 311
006 426
007 564
008 192
008 560
008 569
008 326
008 031

007 877
007 800
007 QIO
008 069
008 215

008 382
008 441
008 487
008 520
008 587

008 699
008 870

009 050
009 185
009 298

009 334
009 324
00g 368
009 532
009 717
010 065
O10 49!
O10 gto
O1T 295
or: 698

O12 045
O12 370
O12 797
O13 334
O13 907

we = =

we

Go
PO KH WO OO O™ An SH HHO OO CHO~? Ann PO HD ww O

>
YO CONT CQwWr

>
ro]

pw pn

wa)
eo Nm O OO COrnTI YN

12
H™,
8 PER-CENT.

Logarithms and Co-logerithms of D,, N,, and M,; with
their Differences.

z | LogD, Colog D Log N, A x

A A
(Log vp) 7 (Colog tps)

an
CONAN PW WHO oO on ak puww 4d

~r

~TI

‘©
re)

Cc
OO DO~ 3 AN PO WV KS

Bo

vw

"094.790

"072 OO
‘048 707
"024 596

"999 631
"973.710
"946 731
"918 585
"889 153
"858 401
"826 280
"792 732
"757 509
"721075

682 593
"641 870
"598 425
551 758
"501 356

"446 989
"389 185
327 504
"261 764
"rg1 821

"£16973
036 279
948 735
"854 268
"752 134

"642 995
527 848
"407 208
"280257
"148 634
"009 003
"853 828
678 14
"477 311
231053} °

95 jo°g10 798

6

"457 823

7 |1°709 032

"977 390

"976 617 /
‘975 889
"975 035

"974 979
"973 O21
"971 854
"970 568
"969 248
"967 879
"956 4.52
"964 937
"963 406
"961 518

"959 277
"959 555
953 333
"949 598
"945 633

"942 196
"938 319
"934 260
"930 057
"925 152
"919 306
"912 456

"799 197
"753 74!

79 746

"905 210

"927 QIO
"951 293
"975 404
"000 369
"026 290
"053 269
"O81 415
‘r10 847

"I4T 599
"173 720
"207 268
"242 331
"278 925

"317 497
°358 130
"400 575
"448 242
"498 644
"553 O11
610 815
"672 496
"738 236
"808 179
°883 027
963 721
"905 533|3°O51 265
"897 866
"890 861
"884 853
"879 450
"872 959
"868 377
"860 369
844 825
824 286/2°146 172
°321 886
"522 68
"768 948
"547 025/1°089 202
"251 209

"145 732
"247 866

"357 005
"472 152
"592 702
"119 743
‘851 366

"99° 997

"542 177
0290 968

"022 700

"023 383
"O24 ILI
024 965
"025 921
"026979
028 146
"029 432
030752
"032 121
"033 548
"035 063
"036 504
"038 482

"040 723
"043 445
"046 667
"050 402
"054 367
'057 804
"061 681
"065 740
"069 9-43
074 848
“080 694
"087 544
"094 467
"102 134
"109 139
"II5 147
"120 550
"127 O41
"131 623
"139 631

"155 475
"175 714
"200 803
"246 259
"320254
"452.975 |0°529 0841179 948

‘748 791 |T'709 032

55 4°16 8591977 93 1/5°883 141) 0°022 069 /5°199 4211964 453

"163 874963 339
"127 2131962 150
"089 3631960 877
050 240959 518

"009 7581958 070
4°967 8281956 524
"924 3521954 879
"879 2311953 126
"832 3571951 250

"783 6071949 231
"732 8381947 048
"679 8861944 673
"624 5591942 064
"566 6231939 216

"505 8391936 120
"441 9591932 792
3747511929 257
"304 008/925 557
"229 5651921 699

"151 2641917 531
"068 795/913 056
3°981 8511908 213
"8go 064/902 908
"792 972/897 152
"690 1241891 036
"581 160/884 729
465 889/878 201
*344 090/871 641
"235 731/864 829
"080 560/857 282

2°937 842/848 343
786 185/837 625

"623 810/822 933)

"446 743/802 793

"249 536/776 657
"026 193/741 417
1°767 610|687 967
"455 577/006 o45
"061 6221467 462

60,

O pw pu

~y
Oo ON AN Pwd KO OO OG NH

co ~~?
WO Dr~T Qn powue

\O
Oo

‘©
“I At - © DS Mm

13
H™,
3 PER-CENT.
Logarithms and Co-logartthms of D,, Nz, and M,; with

their Differences.

Colog M; A x

03§ $47 |3°908 289 | 985 422 g-o91 711 | org 578| 55

oO Com A

n

“vn

~~

Oo
I
2
3
4
5
6
7
8
9
7°
I
2
3
4
5
6
7
8
9

oo co
© GO Ain SO BH =m O

‘oO
oO

433 377

"494 161
"§58 041
"625 249
"695 992
"779 435
848 736
"931 205

4°018 149

109 936

"207 028

"309 876
*418 840
"534111
"655.910
"784 269

"919 440

3°062 158

"213 815
°376 190
"553 257
"750 464

036 661
037 850
039 123
040 482

041 930
043 476
045 121
046 874
048 750

050 769
052 952
055 327
057936
060 784

063 880

"893 711
878 402
"862 308
B45 385
"827 478
"808 468
*788 240
"766 678
"743 651

"719 153
693 156
"665 622
636 478
"605 745

573 O17
"537 879
"499 801
"458 219
"412 567
"362 628
"399 095
"251 506
189 705
"123 590
"052 416
"975 169
"890 772
"799 196
"699 053
"592 923
"480 160
-362 111
°237 687
"109 IIO

"972 883

974 003
972 466
970 856
969 267
967 266
964 868
961 922

"106 289
"121 508
"137 692
"154 O15
"172 522
"IQI 532
"251 760
233 322
"256 349
280 847
°306 844
"334 378
"303 522
"394 255
“426 989
"462 321
"500 199
"941 781
"587 433

"637 372
"690 905
"748 494
"810 295
876 410

"log 228
"200 804
"300 347

O15 309
016 094
016 923
017 907
OIg‘o1o
020 228
O21 562
023 027
024 498

025 997
027 534
029 144
030 733
032 734
035 132
038 078
o4r 582
045.652
049 939

053 533
057 589
061 801
066 115

o71 174

077 247
084 397
ogt 576
099 543
106 730

T12 763
118 049
124 424
128 577
136 227
151714

6

7
8

DO porn mn

Oo cons WD

wy

co
OO OT An FO Kw O OO GAM Hw HW | O

re)

172 206
197 062
242 597
316 787

449 862
746 460

_°973 807
2°232 390
"544 423
‘938 378
1°479 916
0°290 968

802 938| °351 037
757 493| °548 099
683 213] °790 696

550 138|t107 483

253 549) °557 345
0°303 805

393955
532 538

ir F&O bY
vo)
“1 Qtr Rh © BD

we i] ~ i)
OO cn~r Atm SP © WD MH O

&o ©
1O CO~r Qin WB WO WO Or? QU Pwb HO

powers

>

PwWHH OO OO CON Qn

24°148 4
23°995 3
23°8142
23°610 3
23°389 7

23°158 1
22°92
22°686 1
22°457 7
22°2429
22°042 5
21°848 4
21°656 2
23°459 6
21°253 9
21°0379
20°8136
20°582 3
20°347 2
20°10g 0

19°867 4
19°622 7
19°372 6
Ig'1169
18°8549
18-587 1
183141
18°0370
17°755 6
17°468 9
17°1762
16°875 7
16°566 2
16°248 3
15°924 I
15°593 5
15°2596
14°923 4
14°584 7
14°242 3

13°896 3
13°545 2
13°188 1
12°826 4
12°461 5

Az

"267 523
"271981
"277 257
"283 195
‘289 620
296 366
5|°303 256
"310 114
"316 765
"323 024
"328 859
"334 513
"340 IIO
"345 836
"351 327
"358 120
"364 051
"371 389
"378 237
"385 175
"392 232
"399 340
"406 622
"414.072
"421 703

"429 503
"437 452
"445 525
"453 721
"462 072

"470 597
"479 349
"488 364

"497 622
"507 066

"516 693
*§26 420
530211
"540 077
"550 049
"566 127
579 354
"586755
"597 291
"607 916

14

- HM,
3 PER-CENT.
Values of Annuities, and Single and Annual Premiums for

Assurance of a Unit.

7010 638
o10 881
‘O11 173
“OIL 507
"011 875

"O12 268
"012 679
‘O13 093
"O13 504
"013 898
"O14 272
"014 641
"O15 O12
015 398
‘ors 810
016 250
"016 717
"017 208
7017 718
"018 247

7018 795
"O19 364
"019 959,
"020 583
‘O21 239

021 928
"022 649
"023 403
"024 191
"025 O19
025 891
"026 816
027 801
028 851
"029 961

031 138
"032 376
"033 674
"035 939
"036 481
"038005
039 625
"O41 356
"043 199
‘045 160

wn
OO On An

-puwynsd

“A
Cnr

bat f wT
mm POH Mw CO LO CONT A

oc

oO

rw)
tO Bw + O

42)
“Ir win

6
7
8
9
oO»
I
2
3
4
5
6
7
8
9

12°093 8|'
11°7242]°
11°353 3)"
ro"98r 3]°
10°608 3 |°

10°2359
9°865 5
9°497 8
9°133 7
8°774 1
8°417 9
8°064 I
77118
7°360 2
7°007 3
6°656 5
6°3109
59748
576526
5°348 2
5061 4
4°7820

°672 740
"683 530
"694 240
"704 844
“715 310

"725 O91
735 996
"746 259
756 499
766 779

"779 995
*787 062
"796 849
"806 234
"815 100

"823 453
"831 592
"839 463
"847 113
"854 613
"861 872
868 741
875 058
"880 902
"886 174

"891 102
"896 009
"gor 184
"906 629
"913 012
"920 196
"927 558
"935 082
"943 169
"950157
958 780
"965 679
"970 874

"04.7 246
"049 404
"051 824
"054 337
"057 O19
"059 874
062 908
066 132
069 555
073 185
'077 954
"o81 199
"085 661
“ogo 488
"095 761
"107 482
"107 656
"I14 246
‘I2I 190
"128 398

135 851
"143 824
"152 304
"161 382
"171 209

"181 737
"192 773
"203 992
"215 431
226758

‘238 337
"250957
"265 625
282 815
"395 793

"335 846
"372935
"419 537
"483 382
"567 187
677 477

"B19 543
"970 874

H™.

THREE AND' A HALF PER-CENT.,

SS |e | a | SS |

|) | i]
Oo On Ain SW LH & O

SS

&9

&9
mn FOKH HO OO OFAN $O HHO TO CN QW WwW DY = O

as

un >
\O cnr A

& © BH HO

32 017°0
30 695°4
29 422°6
28 197°2
27 O17°9
25 882°3
24 787°7
23 7313
22 711°8
21 729°!
20 782°7
19 872°9
18 999°6
18 160°0
17 359°°7
16 570°2
1581477
15 082'1
14 372°4
13 685°9
13 021°8
12 380°8
11 762°7
{1 165°5
10 587°3

T 556 354
1 488 195
I 422 604
T 359 441
1 298 592

1 239 963
1 183 479
1 129 083
1076 730
1026 389

16
H™

34 PER-CENT.
Commutation Table.

29 720 786
28 164 432
26 676 237
25253 933
23 894 192
22 595 600
21 355638
20172 159
19 043 076
17 966 346

978 031°!
931 603°7
887 048-0
844 293°7
803 264°5
763 886-0
726 091'5
689 819°!
655 015°2
621 629'5
589 612°6
558 917°2
529 494°6
501 297°5
474279°5
448 397°2
423 609°6
399 878°3
377 166°5
355 437°4

334 654°7
314 781°7
295 782°2
277 622°2
260 271'5

243 701°3
227 886°7
212 804°5
198 432°1
184 746°2

171 724°4
159 343°5
147 580°8
136 415°3
125 8281

16939 957
15 961 926
15 030 322
14 143 274
13 298 981

12.495 716
II 731 830
11 005 739
10 315919
9 660 go4

9 039 275
8 449 662
7 89° 745
7 361 250
6859 953

6 385 673
5937 276
5 513 967
5 113 788
4736 622

4381 185
4.046 530
3731 748
3 435 966
3158 344
2 898 072
265437!
2 426 484
2 213 680
2015 248
1 830 501
(658777
1409 434
1 351 853
1215 437

14.715°3
14 552°6
14 375°4
I4 171°4
13. 929°0

13 649°6
13 353°9
13 052°2
12 757°6
12 4781

12 214°9
11 962°7
11 718°6
11 476°6
11 235°4

109957
10 756°8
10 522°0
10 291°6
10 065"9

9 843°87
9 624°46
9 406'32
9 189°33
8974°71
8 763°05
8 55011
8 354°76
8157°71
7 962°48

7 768°74
7573°55
7 37582
717615
6975°64

677438
6 573°70
6 374°29
6 174°85
597419

567 167°5
551 393°3
535 7749
520 508°7
595 453°2

490 5756
475 860°3
461 307°7
446 932°3
432 760°9

418 831°9
405 182°3
391 828'5
378 776'3
3606 018-7

353 54°°5
341 325°6
329 362'°9
317 644°3
306 167°7

204 932°3
283 936°6
273 179°8
262 657°8
252 366°3

242 300'4
232 456°6
222 832°1
213 425'°8
204 236°4

195 261°7
186 498°7
177 942°6
169 587°8
I61 430°1

153 467°6
145 698'9
138 125°3
130 749°5
123 573°3
110 597°7
109 823°3
103 249°6

96 875°33

of Oo &> we ~] | bat
mw Hm O OO Cr Qn -&& bw = O YO CON AN #€$$ O&O HMO OO CON Ati fH YW = O

as
©O cont Qn

3

90700°49| 4

17
H*™,
$1 PERCENT.
Commutation Table——(continued).

Nz S, M; R, z
55 19 027"4 115 800°7 = -{1 089 609 5772°32 |84726°30 55
6] 9484°51 [106 3162 973 808°7 15 568°54 |178953°98 6
7| 895802 | 97 358°14 867 492°6 {5 362°79 = |73 385°44 7
8} 8447°46 | 88910°68 779 134°4 1515516 [68 022°65 8
9| 7952°65 80 9 58°04 681 223.7. |4946°00 |62867°49 9
60] 7472711 | 73 485°93 | 600265°7 14734°49 |57921'49 =| 60
1] 7005.18 | 66480°75 | 5267798 |4520°15 [53 187°08 I
2) 655145 | 5992930 | 460299°0 |4303°30 [48 666'94 2
31) 6110°66 53 818°65 400 369°7. |4084°06 = |44 363°63 3
4| 5 682°67 48 135°97 346 551° [3 862°72 |40 279°57 4
651 5 268°63 42 867°34 298 415°r |3640°84 136416°85 65
6} 486938 | 37997796 | 2555478 (3419°76 (3277601 6
7} 4485°63 | 33512°34 =| 217549°8 |3200°67 $= |29 356°25 7
8} 4117°72 | 29 394°61 184.037°5 |2984'45 |26155°58 8
9] 376669 | 25 627°93 154642°9 |2772°66 |23 17113 9
79} 3430°63 | 22197°30 | 1290149 |2 563°99 |20 398'46 70
I} 3 108°48 19 088°82 106817°6 |2357°84 |17 834'48 I
2| 279898 16 289°84 87 728°81 |2153°46 9 |15 476°63 2
3] 2 501°67 1378818 -| 71 438°97 |1950°80 1/13 323°17 3
4} 221679 | 11571739 | 57650°80 |1750°52 {11372°37 | 4
75| 1946°50 9 624°883 46079°41 11555°20 | 9621°842 [75
6| 1695°69 7929191 | 36454°53 |1370°21 | 8066°643 | 6
7} 1464°08 6465116 | 28 525°34 [1195794 | 6696430 | 7
8} 1252°34 5 212°781 22 060°22 {1 033°71 5 500°491 8
9 | 1060°90 4150°881 16847°'44 | 884°622 | 4466°784 9
80] 888°634 3 263°247 12695°56 | 748°233 | 3 582°162 | 80
1] 734°388 | 2 528°859 9 432°314) 624°037 | 2835°929 | 1
2} 597°419 | 1931440 6.903°455| 511902 | 2209892 | 2
3| 478309 | 1453°131 4972°015| 412°995 | 16977990 | 3
4| 376245 1 076°886 3.518884) 327°106 | 1 284°995 4
85} 291°2326 785°660 0 2 441°998| 254°809 957°889 5 | 85
6] 222°321 563°339 5 1 656°338] 195°752 703°0802 | 6
7| 167°620 395°719 4, 1092°998) 148°570 | 507°3279 | 7
8| 124°504 271°215 7 697°279| I1I'122 358°7580 | 8
9} 915073] 179°7084 426°063| 82°335 7} 247°6361 | 9
go 66°0270 113°681 4 246°355| 59°9499} 165°300 3 | 90
tT] 4579668) = 67°7146 132673) 42°122 5) 105°3505 | 1
2 30°5229 37°I917 64°959| 28°2331 63°228 oo] 2
3 19°1302 18°061 5 27°767| 17°8725} 34°99495! 3
4 10°798 4 7°263 1 9°706| + 10°1876 17°12241| 4
95 5140 5 2°122 6 2°443| 4°8948) 6°934 83] 95
6 1°802 7 °3199 "320 1°7309 2°040 00
7 "3199 "0Q0 O "000 309 1 °309 09} 7

18

H®™,
33 PERCENT.
Logarithms and Co-logarithms of D,, N,, and M,; with

their Differences.

10 14°850 597/1°982 9206/5" 149 403|0°017 074 |6°192 108/980 552
*1660 .

w -_

we

&
OO cor~1r An b&Ho we O HO OT QA SPW HK OC 6 DW AN OW DH

as wo
me pwwn sd

WN
m&w HHO /O Cr AO

"833 523| 98
"816 846) °
"800 462] °
"784 254] °
768 112) °
"751 923) °
735571) *
"718 941) °
"FOTOS
"684 472
"666 774
648 903
"630 980
"613 092

"595 259
"577 430
"559 576
"541 627
"523 561

"505 380
"487 073
"468 680
"4.50 206
"431652

"413 003
"394 235
"375 321
°356251
"337 042
°317 701
"298 262
278 743
"259 116
"239 316
"219 328
"199 060
"178 463
"157 53°
136273
"114672
"092 749
"070 508
"047 878
"024 784

"979 732
"979 403

‘979 068
"978 742
"978 399

"978 077
977 759
977 379
"976 906
‘976 403

"982 557) °
"982 302) °
"982 129) °
"982 077) °
"982 112} °
"982 167) °

982 171
"982 146
"982.051
"981 934
"981 819

"981 693
981 607
"981 526
"981 446
"981 351
"981 232
981 086
"980 930
‘980 791
"980 659
‘980 561
‘980 48:
‘980 373
"980 200
"980 012

"404 741] °
"422 570 °
"440 424] °
"458 373] °
"476 439

*494 620
"512 927
"5351 320
°549 794
"568 348

"386 997
"605 765
"624 679
"643 749
662 958
682 299
"701 738
"721 257
"740 884
"760 684

"780 672
"800 940
"821 537
"842.469
"863 727
"885 328
"907 251
"929 492
"952 122
"975 216

"018 181

"018 307
"018 393
"018 474
"018 554
"018 649

7018 768
"018 O14
"O19 070
‘OIQ 209
"O19 341

"019 439
"OTQ BIO
"O19 627
‘org 800
“019 988

"020 268
020 597
"020 932
"021 258
"021 O01

"021 923
"022 241
"022 630
"023 094
"023 597

5°99° 3531978 878
"969 2311978 716

"793 5321977 935

"779 5671976 781
‘747 3481976 514

"651 6631975 303
‘626 9661974 962

234 8321967 502
"202 3341966 696
"169 0301965 833
"134 8631964 915
"099 7781963 933

ew = a
YO COnT Aw #-& DP = CO

be

&o <)
CO ON AN PW HHO OHO OO Qn $0 DN O

as

->G& YN HO © arr ah - & DS

ond

°
I
2
3
4

15
6
7
8

9

Colog N,

7°807 892
"827 340
*846 916
"866 640
886 527

"906 591
"926 839
947 274
"967 893
°988 688

20 |6'009 647

te)

fe > Go 9
&ONm OO HMO Cr Qin F&O YM OO CO GD~™ Qi PO HD

iw
PWN HO OO CON? Qn

"030 769
052053
°073 506
"095 141
"116.971
"139 009
"161 265

"183 749
"206 468

"229 433
"252652
276 138
"299 904
"323 966

"348 337
°373 034
"398 072
423 467
"440 237

"475 403
"501 Qgo
"529 028
"559 546
"584 573
"613 142
"642 281
, 672 019
"702 388
"733 424
"705 168
797 666
"830970
"865 137
"900 222

A

O19 448 |4°200 420
“TOT 133
"183 723
"177 696
"172 533
"167 768
"162 942
"157 621
"151413
"143 920
"135 119
"125 607
"115 684
"105 768
"096 150
"086 892
077 829
7068 875
"059 Bt4
"059 591
"O41 222
"031 682
"022 097
"O12 482

‘002 851

3°993 166
"983 377
"973 420
963 284
"953 920
"942 656
"932 277
"921 934
‘911 568
"901 048

"890 351
"879 299
"867 811
"855 892
"843 584
*830 870
*817 810
"B04 432
"790 626

"776 279 |

019 576
O19 724
019 887
020 064

020 248
020 435
020 619
020 795
020959

O21 122
o21 284
O21 453
021 635
021 830

022 038
022 256
022 484
022 719
022 965

023 219
023 486
023 766
024 062

024 371
024 697
025 038
025 395

025770
026 166

026 587
027 038
027 518
028 027
028 569

029 139
029 738
030 369
031 036
031 744

032 498
033 304
034 167
035 085

036 067 |

19
H™,

33 PER-CENT.
Logarithms and Co-logarsihans of D,, Nz, and M,; with

x Differences.

Log M,;

A

992 590
993 973
994 837
995 235

995 174
994 679
993 792
992 597
99F 19g

990 488
99° 977
990 084
990 382
99° 742

99° 937
991 046
99° 939
99° 777
ggo 631

990 460
990 415
990 385
990 369
9GO 315

QQO 211
990 043
989 864
989 736
989 636
989 621
989 657
989 634
989 480
989 303
988 948
988 512
988 081
987 692
987 286
986 940
986 622
986 194
985 653
985 072

Colog M,

99° 713 5 799 580

8 867

‘816 277
822 304
"827 467
"832 232
"837058
"842 379
"848 587
*856 0Bo

"864 881
"874 393
884 316
"Bo4 232
"903 850
"913 108
922 171
"O31 125
"940 186
"949 499
"958 778
968 318
"977 903
"987 518
"997 149
4006 834
7016 623
"026 580
"036 716
046 980

"O57 344
"067 723
7078 066
"088 432
"098 952
"109 649
"120 701
"132 189
"144 108
"156416

"169 130
°182 190
"195 568
"209 374
"223 721

A

009 287
007 410
006 027
005 163
004 765

004 826
005 321
006 208

007 493
008 801

009 512
009 923
009 916
009 618
009 258
009 063
008 954
009 061
009 223
009 369

009 540
009 585
009 615
009 631
00g 685

009 789
009 957
O10 136
010 264
O10 364

O10 379
O10 343
O10 366
O10 520
010 697

O11 052
o11 488
O11 QI9
012 308
012 714
013 060
013 378
013 806

O14 347
014.928

x

od &o &> SS) ) mt tn
&eOwON me O OO DOr Ain FW HHO CO CDW) Qin HBO HD KH CO OO CO~ I Qin PW LY wm O

as
\O CONT Aw

iw
oO

ht D -m

20

«i,
34 PERCENT.
Logarithms and Co-logarithme of D,, N,, and M,; with
their Differences.
z | LogD A Colog D A Log N A
* | (Logep,z) 7 (Colog vp) ad
55 |4001 187|1'975 82815-9098 813|0°024 172 |5°063 7111962 888} 55
6 13°977 O15] °975 197|4°022 985) °024 803 | 026 5991961 7731 6
7 | 952212) "974 514) '047 788) °025 486 |4°988 372/960 582] 7
8 | "926 726! °973 786] °073 274) 026214] ‘9489541959 306] 8
9 | ‘900 512] "972 932] ogg 488) °027 068 | -g08 2601957 944] 9
6o | 873 444] 971975) °126 556] °028 025 | -866 2041956 492) 60
I | °845 419) 970918) 154 581) °029 082 | 822 6961954943] 1
2 | *816 337] "969 753) °183 663) 030249 | *777 6391053 294) 2
3 | *786 088) -968 465) 213 912) 031 535 | *7309331951 537] 3
4 | °754 553) "967 145) °245 447) 032.855 | *682 4701949657) 4
65 | °721 698) °965 776] °278 302! °034 224 | °632 1271947 633] 65
6 | °687 474) "964 349] °312 526) "035 651 | *579 7601945 445) 6
7 | 651 823) *962 834) °348177| 037 166] °525 2051943 063| 7
8 | 614.657) "961 302] °385 343] °038 698! -468 268\940 446] 8
9 | 575959} 959415] 424.041] 040 585 | «408 7141937 586, 9
72 | °535.374 "957 174| °464.626) °042 826 | +346 3001934479] 70°
I | °492 548) 954 452) "507 452) "04.5 548 | -280 7791931 138) 1
2 | "447 000] "951 220) °553 000) 048771 |--211 917/927 5901 2
3 | 398 229] 947 496) ‘601 771) 052 504| *139 5071923 879) 3
4 | °345 725} 943 53°} °654275| 056470 | 063 386920009, 4
75 | °289 255] °949 092! °710 745) 059908 |3°983 3951915 834) 75
6 | °229 347| °936216) °770653| °063 784 | -899 229/911 347] 6
7 | °165 563| °932157| 834.437] ‘067 843 | ‘B10 576\906 493] 7
8 | °097 720) "927 954] ‘902 280) °072 046 | °717 o6gi\901 176] 8
Q | °025 674) "923 049} °974 326) °076951 | 618 2451895 405] 9
80 |2°948 723| °917 203/3°051 277| °082 797 | °513 6501889 275) Bo
I | *865 926] ‘910 353] °134074) ‘089 647 | 402 925/882 956, 1
2 | °776279) °903 430} °223 721] °096 570 | *285 881/876424) 2
3 | 679 709| °895 762) 320291] *104 238 | +162 305/869 865) 3
4 | °575471| ‘888 759] -424 520] 111241 | -032 170/863 065} 4
85 | 464230) “882 749) “535 770) °117 251 |2°895 2351855 535] 85
6 | °345979| 877 347] °653 021] °122653 | -750770/846617| 6
7 | 224 326) °870 856] °775 674] °129 144 | °597 387/835 928) 7
8 | ‘095 182] *866 274! ‘904 818] °133 726} °433 315|821 2531 8
9 |1°961 456) °858 265/2°038 S44) “141 735] °254 568801 121) QO
90 | ‘819 721) 842 723) 180 279) “157 277 | 055 6891774 993) 9°
1 | '662 444] "822 182! °337 556) -177 818 |1°830 682/739 763) 1
2 | °484 626) 797 O94! °515 374) *202.906 | "570 4451686 306, 2
3 | °281 720] °751 638) °718 280] °248 362 | -256 751/604 368) 3
4 | 033 358] “677 643} "966 642) -322 357 [0°86 1191465 751| 4
95 |O°711 COL °544 922/1°288 OQO! °455078 | °326 870/176 159, 95
G | °255 923) ‘249 106} °744077| °750 894 |T'505 029 6
7 0°404 971 7

1°505 029

a

eS -_—-

a ee

984 391 |4°238 649

oY
© CONT ATH

“A
.@]

OP po pm

“J

~r

CO
Oo Ow An AhoOKHH DO HO OAH PO H™ CO OO CON A

CO

pwpns

\O
NAN

A

21

H™,
34 PERCENT.

their Differences.

Log Mz

037 112 (3°761 351

038 227
039 418
040 694
042 056
043 508
045 057
046 706
048 463
05° 343

052 367
054 555

5 | 056937

059 554
062 414
065 521
068 862
072 410
076 121
079 991

084 166
088 653
093 597
098 824
104 595
110725
117 O44
123 576
130 335
136.935

144 465
153 383
164072
178 747
198 879
225 007
260 237
313 694
395 632
534 249

2

I

745 742
"729 391
"712 242

"694 254

675 268
"655 153
*633 802
"611 092
"586 893
"561 202
"533 996
"BOK 241
"474 865
"442 897
*408 916
"372 515
"333 138
"2.90 213
"243 168

"191 786
"136 788
"077 799
"org 398
946 758

"874 037
"795 210
"709 187
"615 945
°514 688
"406 216
"291707
"171931
"045 800
"Q15 588

777 788
"624 514
"$50 758
"252 186
"008 071

821 841 |0°68g 738

"238 275

1°490 088

A

983 649
982 851
982 o12
g81 o15

979 888
978 649
977 290
975 Sor
974 3°90

972 794
971245
969 624
968 032
966 o19

963 599
960 623
957 °75

952 955
948 618

945 002
940921
936 689
932 360

Colog M;

"254.258
°270 609
287 758
"305 746

"324 735
"344 847
366 198
388 go8
"413 107
"438 798
"466 004
"494 759
"525 135
"557 103

"591 084
"627 485
666 862
"709 787
"756 832
"808 214
"863 212
"922 291
"985 602

927 279 |3°053 242

921173
913 977
906 758
898 743
891 528
885 491
880 224
873 869
869 788
862 200

846 726
826 244
801428

755 885
681 667

548 537
251 813

"125 963
"204 790
"290 813
"384.055
"485 312
"593 784

"708 293
828 069

"954200

2°084 412

°222 213
"375 486
"549 242
"747 814
"991 929

I°310 262

"761 725

0°509 912

Logarithms and Co-logarithms of D,, N,, and M,; with

A

O15 609
016 351
O37 149
017 988
o18 989

020 112
021 351
022 710
024 199
025 691
027 206
028 755
030 376
031 968
033 981
036 401
039 377
042 925
047 045
o51 382

054 998
059 979
063 311
067 640
072 721

078 827
086 023
093 242
IOI 257
108 472

114 509
119776
126 131
130 212
137 800

153 274
173 756
198 572
244115
318 333

451 463
748 187

wa
© corr An

co Cc ~ ~ ro “a
© OVI An Hw YH O OO CDT AM AW WwW O LO DWr~ AM BPO HW Hw O

\O-
°

YO
SrA fo BW &

» uo &> . S) ) i" be
0 OO ONAN FO HH O OHO TH Qn LHW HK CO WO DW Qn HS Ww BH O

pw we

‘ ws
P&H me O OO CON NH

21°9539
21°834 2
21°688 9
21°522 8
21°341 2

21°49 3
20°952 6
20°7566
20°566 8
20° 389 I

20'224 6
20°065 8
19°908 7
19°747 5
19°5779
19°398 6
IQ'2IL SF
1Q°O17 7
18°820 2
18'619 6

18°415 6
18°208 5
17°996 2
17°778 3
17°554 2

17°324 5
17°089 5
16°850 3
16°606 6
16°357 6
16°102 6
15°8397
15°56079
15°287 6
15*000 7

14°707 2
T4°409 8
14°109 7
13°806 4
13°499 ©
13°187 4
12°870 2
12546 §
12°217 6
11°884 9

22

H™

34 PER- |
Values of Annuities, and Single and Annual Premtums for

CENT.

Assurance of a Unit.

"223 781
"227 830
"232 743
°238 360
"244 500

"25° 990
"257 043
"264 271
"270 690
"276 698

"282 259
287 629
"292 Odi
"298 392
"304 129

"310 104
"316 519
"323 072
329 751
"339 535
"343 433
°350 437
"357 616
"304 986
"372 562

"380 332
"388 277
"396 368
"404 606
"413 026
"421 652
"430 541
“439 735
"449 213
"458915
°468 838
"4.78 894
"489 044
"499 300
"509 695
"520 232
"53° 959
"541 906
"5.53 929
"564 280

Wx

"009 749
009 978
"O10 258
"O10 583
"010 944

“OI 332
"OL 736
"O12 1471 -
"Ord 551
"O12 936

"013 299
013 654
"O14 O10
"014 382
"014 780

"O15 207
"O15 660
"016 139
"016 637
"O17 153
7017 688
"018 244
"018 826

‘019 437
"020 080

020 755
°041°464
"022 205
‘022 980
023 795

024 654
025 567
026 541
"027 580
028 681

"029 849
"031077
032 366
"033 722
"035 154
"036 669
038 281
"040 003
"041 B40
043 794

at
© Orr AR Aw HD © OIOAN

fo) ~ ~
0 OO Ox Qh — CO'b OO

Go
0 UO On An SO WD

pownsd

m1 OG

az

11548 5|°
11°209 5 | °
10°868 3 | °
10°5251|°
10°180 0|°

9°8347]°
9°490 2|°
9°147 5] °
8°807 3] °
8°470 7 |°
8°136 3|°
7°803 4|°
74701)"
713861"
6°803 8|°

6°470 3|°
6°1409 |"
5°819 9}
57511 OF °
52199)
4°944 71"
46761)"
4°415 8):
4°162 4]°
3°913 5] °
3°672 2]°
3°443 51°
3°233 O}°
3°038 11°
2°862 2]°

2'697 8 °
2°533 9)"
2°360 8|°
2°178 4)"
1963 9)"
17217).
1°473 1)"
1218 5]°
"O44 1)"
6726)"
‘4129]°
W775"

g6o 178
966 184

Wr

"045 875
"048 087
"050 442
"052 950
"05.5 629
"058 480
“O61 510
"064 730
068 148
"071 773

"075 637
079 775
"084 233
"089 055
"094 326
"100 047
"106 222
"112 813
"119 756
"126 958

"134 400
"142 362
"150827
"159 890
"169 703
"180215
"IQI 232
"202 424
"213 828
"225 104
"236 617
"249 156
"263 731
"280 Bro
°303 580

"333 595
"370 531
"416942
"480 553
"564 053
"673 936

"815 464
"966 184

H™.

FOUR PER-CENT.

>» ww & iS) YS) bt bm
BO H He GO WO CONIA BPW HH OC LO DWA7 ANN RO HD WO OO CH~ Qn LW HW m= O

aN

On
&eOokP me O OO Or An

67 550°4
64 639°8
61 905°7
59 327°!
56 879°0

54 549°3
52 292°0
50 117°6
48 002°9
45 935°4

43 914°9
41 958°6
40 073°5
38 268°5
36 547°8

34 908°7
33 343°6
31 846°8
30 410°5
29 O31

27-707°1
26 435°7
25 21777
24.051°2
22 934°6
21 864°9
20 839°5
19 855°5
Word
18 005'9

17 138'9
16 309°8
15 518°1
14 761°!
14 035°4
13 339°6
12 670°2
12,025°2
II 404°3
10 807°3

10 233°5
9 682°92

92 63891

24
H™

4 PERCENT
Commutation Table.

N, 8, . M;
1356297 | 24434671 [12 792'8
12915658 | 23078374 |12.474°5
1229752 |21 786716 |12226°6
1170425 |20556964 |12.029°0
1113546 {19 386540 |11 862°6
1059 005 18272994 [11 711°6
1006 713 17 213988 |11 561°1

956 595°8 | 16207275 |1F 397°8

908 593°0 115250679 |11 210°7

862 657°6 | 14 342 086 |10989°'5

818 742°6 | 13 479 429 [10 735°8

776 784'°0 | 12660686 {10 468'5

736710°5 | 11 883 902 |10197°2

698 442°0 | 11147 192 | 9933°49
661 894°2 |10448 750 | 9 684°60
626985°5 | 9786855 | 9451°27
593 941°9 | 9159870 | 9228°73
561 795°! | 8566228 | gor4'40
53% 384°7 | 8004433 | 8 802°97
502 353°5 | 7473048 | 8 593°27
474646'5 | 6970695 | 8 385°77
448 210°8 | 6496048 | 8 180°02
422993°2 | 6047838 | 7 978°77
398.9419 | 5624844 | 7 782°25
376.007°4 | 5225902 | 7 590°64
354 142°4 | 4849895 | 7 403°12
333 392°9 | 4495753 | 7218°66
313447°4 | 4162450 | 703614
294 536°3 | 3849002 | 6855°46
276 539°4 | 3554466 | 6677°62
2593915 | 3277936 | 6 503°07
2430817 | 3018 544 | 6 333°24
227 563°6 | 2775463 | 6168°78
212 802°5 | 2547899 | 6008°61
198 767°2 | 2335096 | 5 850°68
185 427°5 | 2136329 | 5 694°72
172 757°4 | 1950902 | 5 53834
160 732°2 | 1778144 | 5 380°69
149 327°90 | 1617412 | 5 222°25
138 520°6 | 1468084 | 5 063'91
128 287°2 | 1329 564 | 4905°75
118 604°2 | 1201276 | 4748°80
109 448'9 | 1082672 | 4593°59
100 800°3 | 973 223°3| 4.439°11

872 423°0| 4 284'42

429 295°!
416 502°2
404 027°7°
391 8or-2
37977/2°2

367 909°6
356 198'0
344 636°9
333 239°!
322 028°4
311 og8'8
300 303°1
289 834'5
279 637°3
269 703°8
260 o19'2
250 567°9
241 339°2
232 324'8
223 521°8

214.928°6
206 542°8
198 362°8
190 384°0
182 601°8
175 OI‘!
167 608-0
160 389°3
153 353°2
146 497°7
139 820°!
133 317°0
126 983°8
120 815°0
114 806°4
108 955°7
103 261°0

97 722°69

> &o &9 Sy) ~) lon) =~
0 OO ONAN PO KPH CO YO OH Qn SPO Hew O OO GN Qn HBO WY & O

& GO WD we

>

bhwbHe O OO C~ XY Qn

~

“I

(oe)
mm &@ OG Bb we O LO CM~r Air POH MO OO CO

CO

B © wr Oo

\o
SIA +S DP Mm

25
H™,

4 PER-CENT.

Commutation Table—(continued).

320°620
250°992
193°342
146'887
I1O°2 14
81°470 3
59°590 8

42°791 0
29°647 I
19°591 7
12'2200
6°864 6
3°252 1

1°135
"200 5

84 946°32

7/7 7°5°37
70 898-86
64 513°33
58 526°87

52 93104
47 710°32
42 850°85
38 340°30
34 165°83

30 314°12
26 771°40
23 523°58
20 550°47
17 855°35
15 407°05

13 199°32
II 220°97

779 78471

694 837°8
617 132°6
546 233°8
481 722°4

423 195°6
370 264°5
322 554°4
279 793°6
240 363°3
207 197°4
176 883°3
ISO 119g
126 588°3

-1106 031°9

9 461°258
7 999°431

6 553°366
53777748
4 367°521
3 507°583
2 782°

2178°256
1 681°213
1 278°817

| 88 176°53

72 760°47
59 579° 35
48 34918
38 887'92

30. 978°49
24.425°13
19 047742
14.679'90
I1172°31

4 129°56

497°321
413°264
337°734

59 084°17

54954°61
50 980°63
47 162°98
43 502°34

39 999°08
36 654°29
33 469°18
39 444°91
27 582°47
24 882°62
22 344°99
19 968°20
17 7 50°04
15 687°68
13 777°20
12 015'°65
10 400°50

8 929°809
7 601675
0 413°742
5 361°886
4 438°284
3 634928
2942°972
2 352°895
1855°575
1442°311

|

“n , “WA wn
Nn Fo Bw ODO WO Or~T Qn

~

~~

6
7
8
9
fe)
I
2
3
4
5
6
7
8
9

958°197 3
797°205 3
513°863 5
366°9769
256°763 1
175°292 8

- 115°7020

72°QI1O
43°263 9
23°672 2
11°452 2
4°587 6
3355
"200 5
"000 ©

271435
214°138
166°142
127°123
96°099 3
71-5948
52°848 8
38°3409
26°842 8
17°927 7
11°309 6
6°424 2
3°0757
1083 6
"1927

1 104°577
833°143
619°004
452°863
325°740
229°641
158°046
105°197| go
66°856) 1
40°OI3] 2
22°086) 3
10°776| 4
4°352| 95
1°276| 6

193) 7

oO
Wm PO HW =

Oo Or DH

26

H™, —
4 PERCENT.
arithms and Co-logarithms of D,, N., and M,; with
“9 their Differences.

z | LogD, Colog D,

A
(Log vpz)

4
(Colog vp,

|

we

Go

Go i)
W Bw Oo © Grr An pw Py Ho Oo O73 An Sw KK CO YO CONAN FPO He DO ©) OZ QAunn HW DH

tir

4 | ‘O11 762) °

‘B10 500] °
791731) °
773 253) °
754.952] °
"736 718) °
"718 436) °
"699 990] °
"681 267] °
662 148] °

642 612) °
622 821)
602 858] -
°582 842) °
562 861! °

"542 934! °
°523 012) °
"303 066] °
"483 023) °
462 864) °

"442 599! *
"422 I9O| °
"401 705) °
*381 137| °
"360 491] °
"339 748 °
-318 888] -
297 881] °
276 718! “9
"255 416) °
"233 982! °
"212 450] °
"199 838] °
"169 117| °
"147 225) °
"125 143] °
"102 782) °
"080 092!) °
"057 067! °
"033 717 °
"010 023} °
3°986 006} °
"961 673] °

"936950

974 812

"189 500
"208 269
226 747
"24.5 048
*263 282
"281 564
"300 O10
"318 733
"337 852

"357 388
"377 179
"397 142
"417158
"437 139

"457 066
"476 988
"499 934
"516.977
"537 136
"557 410
577 810
"598 295
618 863
"639 509
"660 252
“681 112
"702 119
"723 282
"744 584
766 018
"787 550
"Bog 162
"830 883
"852775
"874.857
"897 218
"919 908
"942 933
"966 283

"989 977

"063 050
"088 238

018 769
018 478
°018 301
018 234

018 282
"018 446

“OIQ IIQ
"019 536

‘019 791
"019 963
"020 O16
"019 981
"O19 927
"O19 922
"019 946
"020 043
"020 159
"020274

"020 400
"020 485
"020 568
"020 646
"020 743

"020 860
"021 007
"021 163
"O21 302
"O21 434

"O21 532
"021 612
"O21 721
"021 892
"022 082

"022 361
"022 690
"023 025
"023 350| °
023 604
"024 O17
"024 333
"024 723
02.5 188

"111 147/978 671
"089 818'978 525
"068 3431978 365
046 708/978 190
024 898/978 008
"002 906/977 823
‘018 723/5°980 729977 O40
958 3691977 470
"935 8391977 308

"913 1471977 153
"BQO 3001976 997
"867 297/976 833
"844 1301976 659
"820 7891976 469

‘797 2581976 267
"773 5251976 053
749 5781975 831
725 4091975 O01
"701 010/97 5 360

"676 370975 112
"651 4821974 851
"626 3331974 577
"600 9101974 286
"575 1961973 982

"549 1781973 661
"522 8391973 326
"496 1651972 974
"469 1391972 604
‘441 7431072 213

"413 9501971 796
"385 7521971 351
"357 1031970 874
"327 977/970 368
‘298 3.451969 829
268 1741969 263
‘237 4371968 666
"206 1031968 038

"141 5151966 668

"108 183/965 917
"074 100/965 112
"039 2121964 250
"003 462/963 331
"02 5 68914°966 7931062 352

iS) om a)

ee

&
&& We O WO CHr AWN POW HO OO CGO™ AM HO WO = OC

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tn
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10 |7°867 645

we

we

wo
YO Cet An $OOv8 mm ODO HO Or? Qn HO BHO } Cn~

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oO pownd

nA
Ow O WO CONT A

Colog N,

"888 853
"910 182
"931057
"953 292

°97§ 102

—997 994
6°019 271

"041 631
"064 161

086 853
"10g 700
"132703
"155 870
"179 211
"202 742
"226 475
"250 422
"274 591
"298 990
323 630
"348 518
"373 067
"399 O9O
"424 804
"4.50 822
477 101
"503 835
530 861
"558 257
*586 044
"634.248
"642 897
"672 023
"701 655
731 826
"762 563
"793 897
825 859
"858 485
"891 817
"925 goo
"960 788
"996 538

5°033 207

O21 329
021 475
021 635
o21 810

O21 992
022 1977
022 360
022 530

022 692
022 847
023 003
023 167
023 341
023 531

023 733
023 947
024 169

024 399
024 640

27
H™.

4 PER-CENT.
Logarithms and Co-logarttihms of D,, N,, and M,; with

their Differences.

Log M,

021 208 |4°106 966
"096 024
"087 304
“080 228
7074 182

"068 617
"062 998
056 821
"049 634
"049 979
“030 834
‘019 886
008 482

3°997 102
"986 082

"975 490
"965 142
‘954.937
"944 629
"934.758

"923 543
"912755
"QO 936
"891 105
*880 279

"869 415
"858 457
"847 335
"836 037
"824 622

813 119
"801 626
"79° 199

778 774
°767 206

"755 472
743 379
°730 838
°717 858
"704 486

"690 705
676 584
662 153
"O47 295
631 892

A

989 058 |5°893 034
"903 976
"912 606
"919772
"925 818

"931 383
"937 002
"943 179
950 366
"959 O21
"969 166
"980 114
‘991 518
4002 898
013 918

991 280
992 924
993 954
994 435

994 381
993 823
992 813
991 345
989 855

989 052
988 596
988 620
988 g8o
989 408

989 652
989 795
989 692
989 529
989 385
989 212
989 181
989 169
989 174
989 136
989 042
988 878
988 702
988 585
988 497

988 507
988 573
988 575
988 432
988 266

987 907
987 459
987 020 |
986 628 |
986 219

985 879

985 569 |
985 142:
984 597!

Colog M,

"O24. 510
"034 858
"04,5 063
"055 371
"06.5 842

"076.457
"087 245
098 064
"108 895
"I1Q 721

"130 585
"141 543
"152 665
°163 963
"175 378
*186 881
"198 374
"209 801
"221 226
"232 794
"244 528
"256 621
"269 162

"282 142
"295 514

"309 295
"323 416

"337 847

"352 795

984012 | 368 108

A

O10 942
008 720
007 076
006 046
005 565
005 619
006 177
007 187
008 655
O10 145

010 948
OII 404
or! 380
O1r 020
O10 592

o10 348
O10 205
010 308
O10 47!
O10 615

oro 788
O10 819
010 831
010 826
010 864

010958
O11 122
O11 298
OFL 415
OFT 503

OTT 493
O11 427
O11 425
O11 568
Or! 734

012 093
O12 541
012 980

O13 372
013 781

O14 121
O14 431
014 858

O15 403
015 988

> > > Go be wo bt boa
0 OO ONAN POH = CO W/O COAT AN Pwo ero © co~r Anni $b =O

Po bw me O WO C* An PO WD

ww

z

55
6
7
8
9

“WA
PB wO DH + O

“A

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~~

CO co
h& G& Bw - O

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6

Log D,

A
(Log vpz)

28

Hm.

Colog D,

4 PER-CENT.

Logarithms and Co-logarithms of D,, Nz, and M,; with
their Differences.

A
(Colog vpz)

Log Nz 4A

3°886 073|1°973 734/4°113 927/0°026 266 |4'929 145/961 305

to

bag

oO

"859 807
832912
"805 333
777 925

"747 864
"717 747
"686 572
"654 229
"620 601

"585 653
"549 337
"51E 593
"472 334
"431 543
"388 865
"343 945
"296 305
"245 441
"190 843
"132 281
"070 280
"OOF 403
"934 467
"860 328
“781 284
"696 394
"604 654
"595 991
"399 660
°286 326
"166 982
"042 236
"910 999
775 180

"631 352
"°471 982
292 071
"087 072
"836617

"512 167
"054.996

7 |1°302 009

"973 105
"972421
"971 692
"970 839
"969 883
"968 825
"967 657
"966 372
"965 052
"963 684
-962 256
"960 741
"959 209
957 322

"955 080
"952 360
"949 136
945 402
"941 438

"937 999
"934 123
"930 064] °
"925 861)3°065 533
"920 956
"915 110
"908 260
"QO! 337
"893 669
"886 666

"880 656
875 254
"868 763
"864 18112°089 oo1
"856172

‘840 630
"820 089
"795 OO1
"749 545|_
675 5591163 383
"542 829) 487 833
"247 O13] "945 004

0°697 991

"140 193
"167 088
"194 667
222.975
"252 136
282 253
"313 428
"345 775
"379 399
"414 347
"4.50 063
"488 407
°527 666
"568 457
"OI1 135
"656055
"703 695)
754 559
"Bog 157
"867 719
"929 720

995 597
"139 672

*218 716
*303 606
"395 346
“$94 009
"600 340
"713 674
833 018
"957 764

"224 820
*368 648
"528 018

"797 929
"912.928

026 895

"027 579
"028 308
029 161

"O30 117
‘O31 175
"032 343
"033 628
"034.948
"036 316
"037 744
"039 259
"040 791
"042 678

O44.920
"047 6.40
"050 864
"054 598
058 562
"062 OO!
"065 877
"069 936
"O74 139
"079 044
"084 890
"OOI 740
098 663
"106 331
“113 334

"ITO 344
"124 746
"131 237
"135 819
"143 828

"159 370
"179 gi! I
"204 999
"250.455
324 450

"457 171
"752 987

"890 450/960 189
"850 6391958 997
"809 6361957 719
"767 3551956 355

"723 7101954 901
"678 6111953 349
631 9601951 696
"583 650/949 936
"533 5921948 053
481 6451946 026
"427 6711943 832
"371 5031941 445
"312 948/938 821
"251 7091935 95"
"187 7201932 832
120 5521929 478
"050 0301925 919
3°975 9491922 196
898 1451918 319

"816 464/914 134
°730 5981909 637
‘640 23 5/904 773
"545 008/899 443
"444 4511893 658
"338 1091887 514
225 623/881 186
°106 809/874 646
2°981 455/868 og1
849 546/861 302

"710 848853 791
"564 639|844 894
"409 533/834 231
"243 704/819 577
"063 3411799 452
1°862 793/773 332
636 12.5)738 114
“+374 239/684 650
058 889/602 694
0°661 5831464 045

"125 6281176 381),

1°302 Q0g

a | x
OW COn~? Qu

60

O pow nm

roe) ~ ~r
Rw H eH O OO CONT A

BWbH m= OC © COX AN

fee)
WO cr Qu

‘Oo
re)

\O
“J Qn RPG H

Logarithms and Corlogaritime of D,, Nz, and M,; with

z | Colog N,

55 5°070 855

"109 550
149 301
"190 364
"232 645
"276 290
"321 389
*368 040
"4.16 3.44
°466 408

"518 3.55
"572 329
628 497
687 052
"748 231
"812 280
"879 448
"949970
4°024 051
"101 855

"183 536
"269 402
"359 765
"454 992
"555 549
661 891
"774377
"893 191
3°018 545
"150 454
289 152
"435 361
"590 407
"756 236
"936.659
2°137 207
"303 875
625 761
‘O41 1T1
1°338 417

874 372
0°697 991

OO pw end © Ord

~T ~
© ON am FWY HO WO COA

‘oO roe) roe)
0 OO ON AN SO HD & O

\O
“IT Qn © bP mm

A

038 695 |3°615 904

039 811
041 003
042 281
043 645
045 099
046 651
048 304
050 064
051947
053 974
056 168
058 555
061 179
064 049
067 168
N70 522
074 081
077 804
081 681

085 866
090 363
095 227
100 557
106342

112 486
118 814
125 354
131 909
138 698

146 209
155 106
165 769
180 423
200 548
226 668
261 886
315 350
397 306

29

H™,
4 PERCENT.

,

their Differences.

Log M,

"599 226
581 796
"563 558
"544 472

"524 369
"503 124
"480 621
"456 736
"431 338

"404 430
"375991
"345 992
"314 364
"281 143
'24.5 897
208 212
"167 521
"123 242
074 792

"021 957

2°965 485

"904 908
"840 079
"770 908
"696 637
"616 227
"528 575
"433 665
"330694

"220 478
"104 223

1°982 720

"854 882
"723 036
"583 663
428 828
"253 524
"053 445

535 955 |0°807 816

823 619

"487 O41
"034 883

7284975

A

Colog M,

983 322 |4°384 096

982 570
g81 762
980 914
979 897

978 755
977 497
976 115
974 602
973 092
971 561
970 OO!
968 372
966 779
964 754

962 315
959 309
95572)
951 550
947 165

943 528
939 423
935171
930 829
925 729

919 59°
g12 348
go5 ogo
897 029
889 784

883 745
878 497
872 162 |2
868 154
860 627

845 165
824 696
799 921
754371
680 125

546 942
250 092

"400 774
°418 204
"436 442
"455 528
"475 631
"496 876
"519 379
"543 204
‘368 662

"595 57°
"624 009
654 008
685 636
718 857

"754 103
"791 788
"832 479
876758
"925 208

"978 043
3°034 515
"095 O92
"159 921
°229 092

"303 363
"383 773
"471 425
"566 335
"669 306

779 522
895777

2°017 280

"1451718

276 964

"416 337
"S71 172
"746 476
"946 555
I'192 184

°§ 12 059
"965 117
O°715 025

4

016 678
O17 430
018 238
o19g 086
020 103

021 245
022 503
023 885
025 398
026 908

028 439

029 999
031 628

033 221
035 246
037 685
040 691
044 279
048 450
052 835

056 472
060 577
064 829
069 17!
O74271
080 410
087 652
094910
102971
110 216

116255
I2I 503
127 838
131 846
139 373

154835,

175 304
200 079
245 629
319875

453 058
749 908

(@, WN
b&O bY HO OO Cr? Qn 8

“nn
wn

“I
Oo Or AN fH We O OO CON A

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co
oO

\o co
bob e O OO Or Qh HPO HD

Wr AK

30

H™,
4 PER-CENT.
Values of Annuities, and Single and Annual Premiums for

Assurance of a Unit.

A; az A;
10 |20°076 5|°189 365 |" T1°042 6} °536 823 | 044 577
I | 19°982 4|°192 985 |° 10°731 1|°548 805 | 046 782
2 | 19°864.9|°197 503 |° 10°416 7 | °560 899 | "049 130
3 | 19°728 3 |*202 757 |° 10°099 6 | °573 093 | "051 632
4 |19°577 5|°208 559|° 9°779 8} 585 392 | °954 395
15 1 19°41609)°214 733 |° Go | 9°459 0) 597 730 | 057 150
6 | 19°251 8)°221 086): I | 9°138 3 | 610 067 | "060 175
7 | 19°087 0/ °227 422]: 2 | 8°818 4|°622 371 | °063 388
8 | 18°927 9|°233 543 1° 3 | 8°500 1 |°634 610] °066 800
Q | 18°779 8 | 239 239 |" 4 | 87184 5|°646751)°070 418
20 | 18°643 8 | °244 468 | - 65 | 7°870 3|°658 834] °074 274
1 | 18°513 | "249 497 |° 6 | 7°5567|°670 894 | 078 405
2 | 18°3840)°254 463] ° 7 | 7°242 9] °682 966 | °082 855
3 | 18251 11°259 574)" 8 | 6°928 1} °695 073 | 087 672
4 | 181104] °264 985] 9g | 6°610 4) "707 294 | "092 938
25 |17°96071|'270 742 | ° 70 | 6'293 0| "719 502 | 098 657
6 | 17°803 8|°276 777): 1 | 5°978 7) °731 589 |°104 832
7 | 17°640 6|*283 055)" 2 | 5°6719)°743 390) "III 422
8 117°473 7 |°289 472] ° 3} 5°3766|°754 746 |°118 362
9 | 17°303 9 |°296 002 |: - 4 | 5°096.9|°765 506) "125 558
3° | 17°1309] "302 658 |° 76 | 4'832 6|°775 668 | "132 988
I | 16°954 8 | "309 432° 6 | 4°574 2|°785 606 | 140935
|16°773 7|°316 396|° 7 | 4323 5|°795 251 | ‘149 386
3 | 16°587 2 |°323 570|° 8 | 4°078 9 | 804 659 | "158 432
4 | 16°394 8) "330 969|° Q | 3°838 2|°813 917 | "168 229
35 | 16°196 8 | °338 584] - 80 | 3°604 3 |°822 O10 |°178 725
6 | 15°993 8|°346 393|° 1 | 3°382 4/*831 445 | "189 722
7 |15°786 4|°354 368] 2 | 3°1780|°839 307 | ‘200 887
8 | 15-5747 |-362 509 | 3 | 2°988 6|-846 593 |-212 254
9 |15°3577|°370 857 | ° 4 | 2°8176|°853 167 |°223 480
4° |15°1347]°379 434 |" 85 | 2°657 8|°859 316 | °234 927
I | 14°904 0] °388 308 |° 6 | 2°498 4|°865 448 | °247 386
2 | 14°664 4] °397 522|° 7 | 2°329 7|°871935 | °261 867
3 114°416 51°407 058 |: 8 | 2°151 6|°878 784 | °278 836
4 |14°161 9] °416 852) 9 | 1°941 6|°886 862 | °301 489
45 | 13°900 5|°426 903]: 90 | 1°703 9|°896 004 | °331 376
6 113°6350]°437 116)" 1] 1°4593|°905 412 |°368 159
7 |13°366 3|°447451|° 2 | 1°208 3} ‘915 066|°414 380
B | 13°094.0|°457 921° 3. | °9372|°925 493 |°477 756
Q | 12°817 3 | °468 564° 4 | *668 3/°935 835] °560954
50 | 1275360479 383 |° 95 | *4106|°945 745 |°670.437
I | 12°248 8|°490 431 |° 6 1766] 954 746 | 811 438
2 |11°9547| "501 742° 7 961 539 | 961 539
3 | 11°65501°513 2701"
4 |11°3509| "524 964)"

- H™.

FOUR AND A HALF PER-CENT.

© CAT Ah PW Ym O

we

<X)

°
I
2
3
4
5
6
/
8
9
30
I
2
3
4
5
6
7
8
9

puwnd

aN

PO KH mM GO OO Or? Qn

64 392'8

61 317°9
58 443°4
55 7410
53 185'2

5° 754°3
48 429°3
46 193°4
44 032°6
41 934°5

39 898°2
37 938'4
36 060°6
342715
32 §73°9

30964'2
29 434°4
27 978°6
26 588'9
25 261°5

23 994'0
22 783°4
21 629°7
20 530°5
19 483°7
18 486'1
17 5348
16 626°9
15 760°3
14 934°2

14 147°0
13 398°3
12 686°9
12 O10°2
II 365°1

10 750°!

10 161°7
9 598°29
9 059°!
8 543°82
8051°47
7 581°86
7 134743
6 707°39
6 299°17

32
H™

41 PERCENT.
Commutation Table.

1 188 642

I 127 324
1 068 881
1013 140

959 9547
90g 200°3
860 771°0
314 577°6
779 54572
728 610°6
688 712°4
65° 773'9
O14 713°3
580 441°8
547 867°9

516 903°7
487 469°3
459 49°°7
432 9018
407 640°3
383 646°3
360 862°9
339 233'2
318 702°7
299 219°!
280 733°0
263 198'2
246 §71°3
230 810°9
215 876'8
201 729°8
188 331°5
175 O44°7
163 634°5
152 269°3

141 519°3
131 357°6
121 759°3
112 700°2
104. 156°4

20 229 255

19 040 613
17 913 289
16 844 408
15 831 268
14 871 313
13 962 113
13 IOI 342
12 286 764
II 516219

10 787 609
10 098 896

9 448 122
8 833 409
8 252 967

7795 999
7 188 196
6 700 726
6 241 236
5 808 334

5 400 694
5 O17 047
4656 185
4316951
3 998 249

3 699 030
3 418 297
3 155.0098
2908 527
2677716

2 461 839
2 260 110
2071778
1 896 133
1732 499
I 580 230
1438 710
T 397 353
1185 594
1072 893

96 104'g0
88 523°03
81 388°61
74 681'22
68 382°05

968 737°0
872 632°!
784 109°!
702 720°4
628 039°2

10 434°33

10 132°39
9 89829
9 712°65
9 55714

9 416°59
9277°15
9 126°68
8 955°°7
8 753°15
8 522°61
8 280°96
8 036°81
7 800°64
7 578°80

7 37184
7175°39
6 987°09
6 802°23
6 619°76

6 440°07
6 262°75
6 090°13
5.922°38
5 759°60
5 Gor'os
5 445°85
5 293°01
5 142°43
499493

4850°85
4711°33
4 576°88
4 446° 56
4 318°68

4 192°99
4.067°57
3 941°74
3 815°88
3 690°70

3 566°27
3 443°37
3 322°43
3 202°61

3 083°23

327 Q6o'r

317 525'8
397 393°4
297 49571
287 782°5

278 225°4
268 808°8
259 531°6
250 404°9
241 449°9
232 696°7
224 174°!
215 893°2
207 856°3
200 055°7
192 476°9
185 105°]
177 929°7
170 942'6
164 140°4
157 520°6
151 080'5
144 817°8
138 727°6
132 805°3

127 045°7
121 444'6
115 9988
110 705°8
105 563°3
100 568'4

95 717°54
Q1 006°21
86 429°33
8 982°77

77 664°09
73 471'10
69 403°53
65 461°79
61 645°91

57 955°20
54 388'9.
5° 945°50
47 623°14
44 420°52

~_ Go &o ) 1S) oo bat
OO O~3 An POKD MO OO COO~N Qh PO Ve OC WO CHr~3 Qn hw W HO

-& BH O

fo

hove EO WO CN QnHn

Oo Cnt An

oO pone

D,;

5 908'95

5 535°57
5 178°26

836°39
4 5°9°53
4196°50
3 896°61
3 609°35
3 334°29
3 071°09
2 820°08
2 58144
2 355°24
2 141°38

H™,

41 PER-CENT.

Commutation Table—(continued).

Nz

62 473°10

56 937°53
51 759°28
46 922°89
42 413°36
38 216°86
34 320°25
30 710°90
27 376°61
24 305°52
21 485°44
18 g04°09
16 548°77

Sz

559 957°2

497 1841
440 246°5
388 487°3
341 564°4
299 151°0
260 934'2
226 613°9
195 903°0
168 526°4

144 220°9

122 735°4
103 831°4

41 337°29

38 373°02
35 527°69
32 801'29
30-193°76

27 704°84

125 33475

23 083°85
20 952°40
18.940" 59
17 048°40
1527497
13 618°74
12077°55

OO Cm Qn

Lo,
.@)

“

87 282°66
72 875°28

60 407°97

49 690°74
40 544°08

I 407°39

12 467°31

IO 717°23
9 146°663
7 7467005

10 648°80

9 329°138
8 115°927
7 006'870
6 000°088

I 940°08
I 750°08

157957
I 400°66

0 ©) O™ Ann LW DH we

~T

125325.
1 109°06
1 006'78

a i

~~

6
7
8
9
fo)
I
2
3
4
5
6
/
8
9

co
oO

Le )
Oo CONT Qn SO Db &

Ve)
re)

“Oo
“SOM & © DB m

1 239°90
1 08819

946°366
816°536
698°258
591°558
496°335

411764
337°935
271551
215°331
167°761

128°610
97°240 8
72°613 8
53°419 4
38°886 3
27°789 9
19°1617
12°602 0
7°8227
4°3734
2°062 0
"7162
"1259

906°337
808023

713°059
623°981
540°865
464°233
394°484
331°286
274°289
223°319
178°792
140°495
108568
82°7368
62°297 2
46°229 8
33°997 ©

24°575 1
17143 6
TI°409 I
71724
4°0600

19369

"6799
"120 5

5 993°75!
4285°728
3 572°669
2 948°688
2 407°823
1 943°59°
I 549°106
1 217°820
943°531
720°213
540421
$00°926
292°358
209°622
147°324
101°095

67°097
42°522
25°379
13°970
6°797
2°737
“800
"I27

~~]
© COn~7 Qin HP © HD mw

OO
oO

\O CO
-t Ow & O Oo ON AN © D

\O
“FAN

34

‘HM,
41 PER-CENT.
artthms and Co-logarithms of D,, N,, and M,; with
“<9 their Differences.

I
2
3
4
15 020 365 15
6 | °685 108 314 892] ‘020 528 6
7 | °664 580 335 420) °020 806| ° 7
8 | 643774 356 226) ‘021 202 8
g | 622 572 377 428] °021 619] ° 9
20 | 600953 399 047| °021 874) ° 20
1 | °579979 420921} *022 046] ° I
2 | °557 033 442907] °022 099) °7 a
3 | 534934 405 066] °022 064| ° 3
4 | °512 870 487 130] °022 010 4
25 | ‘490 860 "509 140] °022 005 25
6 | *468 855 "531 145| 022029] ° 7 6
7 | 446 826 553.174, °022 125 1]
8 | °424 701 575 299| °022 242 8
Q | 402459 597 541| °022 357| “61027 9
30 | °380 102 619 898] °022 483 30
I | °357 619 642 381| *022 569 I
2 | °335 95° 664.950] 022650 2
3 | °312 400 687 Goo] *022 730] ° 3
4 | °289 670 710 330] °022 825 4
35 | °266 845 733,155] °022943 35
6 | °243 902 756 098] °023 091 6
7 | °220811 "779 189] °023 245 7
8 | *197 566 "B02 434! °023 385 8
9 | 174.181 "825 819) *023 517 9
40 | °150 664 "849 336) °023 615] ° 7 40
I | °127 049 872951) °023 695 I
2 | ‘103 354 896 646] *023 804 2
3 | °©79 55° "920 450) °023 975 3
4 | 955 575 944.425) "024 165 4
45 | “O31 410 45
6 | 7006 966) °975 228] -993 034] °024 772 6
7 |3°982 194) 974 89114017 806] 025 109| ° 7
8 | "957 085] °974 567| 042 915] °025 433 8
9 | 931 652 "068 348) °025 777 9
50 | "905 875 "094 125] 026 099 14982 7461964 310! 50
1 | 879776 "120 224| °026417] '947 0561963 508] 1
2) *853 359 "146 641| °026 806] ‘g10 5641962 647| 2
3 | 826 553 "173 447| °027 270) °873 2111961 731) 3
4 | °799 283 "200 717] '027 772} 8349421960751] 4

&

) o~ ma
Oo ON Aw PO DH HO

ww

&o

aa > oo
BoP me O O ONAN FWHM O OHO ON ON #$& Dm O

oY
PO De QO OO CONT AN

Colog Nz

7924949
"947 951
"971 O71

_ 994 331

6°017 749
"O41 340
"065 112
"089 068
"113 202
"137 504
"161 962
"186 570
‘211 327
"236 241
"261 324
°286 590
°3.12 053
"337 723
"363 611
"389 723
“416 069
"442 658
"$09 502
"496 O14
"524 OLT

"552797
"579 707
608 057
"036 744
"065 794
"695 230
725977
"755 395
786 125
"817 387

"849 184
"881 545
"914 498
"948 075
982 314

SOL] 254
"052 944
"089 436
"126 789
"165.058

35

HM,
41 PERCENT,

Logarithms and Co-logarithms of D,, Nz, and M;; worth
their Differences.

023 002 14'018 465

023 120
023 260
023 418
023 591

023 772
023 956

024 134

024 302
024 458
024 608
024757
024 O14
025 083
025 266

025 463
025 670
025 888
026112
026 346

026 589
026 844
027 112
027 397
027 696
028 010
028 340
028 687
029 050
029 436
029 847
030 288
030 760
031 262
031 797
032 361
032 953
033 577
034 239
034940

035 690
036492
037 353
038 269
039 249

"005 712
3°995 560
"987 338
"980 328

"973 894
"967°414
"960 313
"952 069
942 164
"939 572
"918 081
"905 084
"892 130
879 601

"867 576
855 846
844 296
"832 652
"820 842

808 890
"796 765
"784 627
772 496
"760 393
"748 270
°730 066
°723 793
"711 169
"698 529
685 818
‘673 144
"660 570
"648 024
"635 351
"622 524
"609 335
"595 688
"581 595
"567 109

"552 214
"536 984
"521 456
"505 595
"489 006

987 247 |5°981 535

989 848] "994 288
991 778 |4°004 440

992 990
993 566

993 520
992 899
991 756
990 095
988 408

987 509
987 003
987 046
987 471
987 975
988 270
988 450
988 356
988 190
988 048

987 875
987 862
987 869
987 897
987 877

987 796
987 637
987 466
987 360
987 289

987 326
987 426
987 454
987 327
987 173
986 811
986 353
985 907
985 514
985 105

984770
984 472
984 049
983 501

982 912

"012 662

"019 672

"026 106
032 586
"039 687
"047 931
"057 836
"069 428
081 919
"094 916
"107 870
"120 399

"132424
"144 154
"155 704
"167 348
"179 158
"IOI IFO
"203 235
"215 373
"227 504
"239 607

"251730
"263 934
"276 297
288 831
"301 471
"314 182
°326 856
°339 430
"351976
"364 649
377 476
"390 665
"404 312
*418 405
"432 891
"447 786
463 016
"478 544
°494 495
"510 O04

A

012 753

O10 152
008 222
007 O10

006 434

006 480
007 101
008 244
009 905
OLl 592

O12 491
012 997
012 954
O12 529
012 025

OII 730
OrI §50
O11 644
o11r 810

O11 952

O12 125
012 138
012 131
O12 103
012 123

O12 204
O12 363
O12 534
O12 640
O12 711

012674
O12 574
O12 546
012673
O12 827

013 189
013 647
O14 093
014 486
014 895
O15 230
O15 528
015951

016 499
017 088

&

bh
© YH O

bm

ee
&Oob em O OO CO™ An

s

Go
Oo cor~7V An SPW HHO WO CONT AN

&o

>
Pe GO Yb & O

aN

PW PHO ©) CONT AN

36

H™,

41 PERCENT.

Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

Log D, Colog D,

A
(Log tp,)

co ~ ~ ro. nv on
Bo ws ae pwn nd O OYAN FHOKHHO OBVAN AW HHO oo 08 | 8

Ve)
“IS AQn $+ © Db

(Coleg tpr)

-028 978
"029 662
"030 391
"O31 244
"032 200
033 258
"034 425
035712
037 030
"038 400
039 827
"O41 342
"042 874
"044 761
"O47 002
"049 724 |3°961 263927 815
"889 0781924 243
"813 321/\920 511
°733 832/916 626

"650 458/912 431
"562 8891907 927
"470 816/903 052
373 868/897 710

"052 946
"056 681
“060 646

"064 083
067 gCo
072 O19
"076 222
‘o81 127
"086 973
093 823
"100 746
"108 414
"115 417
"I21 426
"126 830
"133 319
"137 993
"145 910

"161 454

"181 993
"207 082

2°9

Log Nz

"713 9881957 397
671 3851956118

"627 5031954752

"582 2551953 295
535 55°
"487 293
"437 380
"385 7951946 43
"332 144944 410
2.76 554,942 212
"218 766)
"158 585\9.
095 7731934310
"030 083

"055 149/817 903
"873 0521797 786
670 838)771 676
"442 514/736 468
*178 982/682 998
"252 539 |0°'861 g8o|601 027
"326 533| *463 007/462 345
"459 254 |1°925 352/174 O10

755070| 099 962

Ga

OO CO™ Aw

951743
950 087

948 325

Pw Pb

“A

OO O~I AW +#wndsre CO OO Or? AN

939 BIg
7 188

931 180

~

~~

pope Dd OOIAS WKH

‘XO
“wT AN

37

H™,
4} PERCENT.
Logarithms and Co-logarithms of D,, Nz, and M,; ‘with
: their Differences.
A Log M; A Colog M;, A

040 204 13°471 918 | 982 216 |4°528 082

o17 784) 5!

“J “nn
kev e QO OO CON QAM PO HD

“sr

co
OO GOsm1 An HON HO DO On An

©O

"244 OO1
°286 o12
328 615

"372 497

“417745
"404 450
"512 797
562 620
"614 295
667 856
‘723 4406
"781 234
"B41 415
"904 227

969 917
4°038 737
"110 922
°186 679
"266 168

349 542
437 111

"329 162
"557 486

821018

Of 411
042 603
043 882
045 248

046 705
048 257
049 913
051675
053 561

055 599
057 788

060 181
062 812
065 6go

068 820
072 185
075757
079 489
083 374

087 569
092 073
096 948
102 290
108 088

114 246
120 584
127 129
133 682
140 459
147 952
156 827
167 462
182 097
202 214

228 324

263 532
317 002

"454 134
"435 590
"416 228
"396 o12

"374 764
"352 358
.328 674
"303 587
"276.965
248 815
"219 120
"187 856
"154.956
"120 464

"083 936
"044 954
"002 936
"957 290
"907 424

"853 125
"795 172
"733 089
666 736
*§96 030
"520 203
"438 208
"348 925
"252 347
"147 661

"035 /Ol
‘97 699
794 469
"664 922
"531 441
"390 495
°234 102
057 25°

398 973 |0°855 667

537 955
825 390

"608 525
"287 109

1°832 451
"080 8.46

981 456
980 638
979 784
978 752

977 594
976 316
974913
973 378
971 850
97° 305
968 736
967 100
965 508
963 472
961 018
957 982
054 354

"545 866
"564 410
"583 772
"603 988
"625 236
"647 642
671 326
"696 413
"723 035
"751 185
"780 880
"812144
B45 044
"879 536
"916 064
955 046

"997 064

950 134 |3°042 710

945 791

942 047
937917
933 947
929 204
924 173
918 005
910717
903 422
895 314
888 o4o

881 998
876770
870 453
866 519
859 054
843 607
823 148
798 417

"0g2 576
"146 875
"204 828
*266 911
"333 204
"403 970

479 797
"561 792
"651075
747 653
"852 339

"964 299

2'082 301

"205 53!
"335078
"468 559
"609 505
"765 898
"942 750

752 858 |1°144 333

678 584
545 342

"391 475
"712 8g!

248 395 |0°167 549

"919 154

018 544
O19 362
020 216
021 248

022 406
023 684
025 087
026 622
028 150

029 695
O31 264
032 goo
034 492
036 528
038 982
042 018
045 646
049 866
054 299
057953
062 083
066 353
070 706
075 827
o81 995
089 283
096 578
104 686
113 960

118 602
123 230
129 547
133 481
140 946
156 393
176 852
201 583
247 142
321 416

454 658
751605

5
6
7
8
9

“
oO

BQ pw So

~
&O bw O ) Con NA

~I

co
CO CO~ YT CQ Fw B MO LO Or? Qn

oe)

Ve)
PwenH 6

\O
“sr Ah

> eS) &> i] wy a] a]
0 OO ON AM FW D-H O © GWT AN SPhOK KH EC OO CG~ I Qn Pw Hb -& O

dO DN

fs.

ir
POH AH CO /CO CONT QAN

15°523 4
15°357 5
15°186 2
I5°0100
14°829 7
T4°045 0
14°455 2

14°259 6
14°056 4
13°844 6
13°6246
13°3979
13°164 5
12°926 7
12°685 5
12°440 5
12°1908
11°936 3
11°675 6
II"407 9
11°34 2

10°8557

38
H™,

44 PERCENT.
Values of Annutties, and Single and Annual Premiums for

Assurance of a Unit.

"162 042
"165 243
"169 366
"174.246
"179 695

"185 533
"IQ 565
197 575
"203 374
"208 734

"213 609
218 274
222 870
"227 613
"2.32 665
"238 076
"243 776
"249 730
"255 830
"262 050
"268 404
274 882
"281 564
288 467
"295 O12
"302 988
"310 573
"318 340
"326 290
"334.463
"342 890
"351 638

"008 327
"008 524
"008 780
"009 087
"009 433
“009 809
"O10 204
"O10 603

"O10 O94
‘O1I 360

"011 697
"O12 O24
"O12 350
"012 690
"O13 057
"013 456
"013 881
"O14 333
"014 804
"O15 292

"015 798
"016 324
7016 877
017 458
018 072

"018 719
"O19 399
"020 110
"020 856
"021 641

"022 470
"023 355
300 758)"
"370 232°
"379 993 |°
"399 044)"
"400 284 |'
"f10671]°
"421221/°
"431973 |"
"442 934 |"
“454 159°
*465 690] °
‘477 475).
"489 466°

Ww
© cnr~r Qui

~y] ~~]
OS puwrnwnd

Oo cosr An PO HH OC WO CO~ A

fore co
© DO~Tr AH Pw Pw HO

ve}
GO WM O

\©o
“Ir AK

az

10°5726

190°285 8
9995 5
9°7020
9°405 3
9°106 8
8°807 7
8-508 7
8°2106
79143
7°618 7
7°3230
7°026 4
6°728 I
6°426 2

6°123 8
5823 8
5°530 3
5°247 3
4°978 8

4°725 0
4°476 2
4°234 5
3°998 3
3°765 3

3°5387
3°323 3
3°1247
2°940 5
2°7743
2°6189
2°463 7
2°209 3
21254
1'9198
1°686 4
1°4457
1'198 2

"939 3

"6640

"408 4

Az

"501 658
"514,010
"526 509
"539 146
"551926

"504777
"577 658
"59° 534
"603 370
°616 130

"628 859
"O41 591
"654 367
667 211
"680 212

"693 231
"7060 553
"718 792
"739 978
"742 538
753 47°
"764 181
"774 592
"784 764
"794 794

"804 554
813 829
822 381
"830 312
"837 469

"844 163
"850 844
"857 925
"865 411
874 266

"884 318
"894 681
"995 337
"916 874
"928 337
"939 340
i 75 8 e
"956 938

949 333

"043 349
"045 545
"047 884
"050 378
"053 943

055 881
058 898
"062 105
065 508
"069 117

"072 964
"077 086
"o81 527
"086 336
"OgI 596
"097 311
"103 484
"110071
"117 007
"124.195
"131 O11
"139 545
"147979
"157 997
"166 787
"177 266
188 243
"199 380
"210 711
221 886

"233 206
"245 O45
"260 034
°276 892
"299 427
"329 188
"365 817
"411 847
"474 991
"557 885
666 964
"807 418
956.938

H™,

FIVE PER-CENT.

40
H™,

5 PER-CENT.
Commutation Table.

2 D, N, 8, M, R, z
1O |61 391°3 [1047 140°2 116859 782 |8604'11 1252 897°6 | 10
1 |58181'4 | 988958°8 |15 812642 |8 317°61| 244 293°5 I
2 | 55 189'8 | 933 769°0 |14 823 683 |8096°55|235975°9 | 2
3 |52387°3 | 881 381°7 |13 B89 914 | 7 922°07 | 227 879°3 3
4 149 747°2 | 831 634°5 |13 008 533 | 7 776°63 |219957°3 | 4
15 1472474 | 7843871 [12176898 |7 645°78| 212 180°6 | 15
6 | 44 868'4 | 739 518°8 |t1 392 511 | 7 516°59 | 204 534'9 | 6
7 142 593°0 | 696925°7 [10652992 | 7 377°85 | 197 018°3 7
8 |40407°3 | 656518°4 | 9956066 | 7 220°36| 189 640°4 8
9 1382987 | 618 219°7 | 9299 548 | 7 035°95 | 182 420°1 9
20 |36265'4 | 581954°3 | 8681 328 |6826'40/175 3841 | 20
I |34319°9 | 547 634°4 | 8099374 |6607°81/168557°7.| 1
2 |32465°8 | 515 168°5 | 7551740 |6 388°00| 161 949'9 2
3 1307082 | 484.4603 | 7036571 |6176°38)155 5619 | 3
4 {290481 | 4554122 | 6552116 | 5978°55)149385'5 | 4
25 [274812 | 4279311 | 6096699 | 5 794871143 407°0 | 25
6 1259990 | 401932"! | 5668768 | 5 621°35|137 O12°1 6
7 \24595°4 | 3773366 | 5266835 | 5 455°82/131990°8 | 7
8 | 23 262°5 | 354074'1 | 4889 499 | 5 294°09/126 5349 | 8
9 1219959 | 3320782 | 4535425 [5 135°20/121240°9 | 9
30 |20792°7 | 311 285°5 | 4203 347 |4979°48/116105°7 | 30
1 119649°7 | 291 635°8 | 3892061 |4826°55/ 111 126°2 I
2 |18565°8 | 273070°0 | 3600425 | 4678°39| 106 299°6 2
3 |17 5384 | 2555316 | 3327355 [453508 |1o1 6212 | 3
4 116 564°9 | 238966°8 | 3071824 |4396°69! 97 08614] 4
35 |15 641g | 223 324°8 | 2832857 |4262°54] 92 68945) 35
6 1147664 | 208 558°5 | 2609 532 |4131'84| 88426'91| 6
7 |13935°1 | 194623°4 | 2400973 |4003°74| 84295°07| 7
8 }13145°9 | 18147775 | 2206350 |3878°14| 80-295°33] 8
9 |12397°5 | 16goBoro | 2024873 (3 755°69| 7641319] 9
40 [11 6881 | 157391'9 | 1855793 |3636°66| 72657750! 40
1 }11016'8 | 14637571 | 1698401 | 3 521°94} 69020°84| 1
2 |10382°2 | 1359930 | 1552026 |3 41191] 6549890] 2
3 | 978163) 126211°3 | 1416033 |3305°77| 62086'99| 3
4 | 9212°18] 1169992 | 1289821 | 320212] 58781'22| 4
45 | 8672°11) 108 327°0 | 1172822 |3100°72] 55 579°10| 45
6 | 8158°46) 100168°6 | 1064495 |3000°03| 52478°38| 6
7 | 7669°41] 92499°17) 964 326'5|2899°48| 4947835) 7
8 | 7204°12| 85295°05| 871 827°3/2799°40| 4657887) 8
9 | 976199} 78533°06| 786 532°3) 2 700°33) 43779°47| 9
50 | 6341°98} 72191708] 707 999'2| 2 602°31] 41079°14| 50
I | 5943°64| 66247°44| 635 808°2/2 505°97| 38476°83| 1
2| 550625, 6068r'I9) 569 560°7/2 411°61| 35970°86| 2
31 520816] 55473°04| 508 879°5]/2 318°58| 33559°25| 3
4 | 4867°89|. 50605'15| 453 406°5|2226°31| 31240°68| 4

60

O ONT AN PWR AO OO DW~AN RYH HRO © wr AS pw we 4

~TS

“I

‘e \e ¢) CO
oO

©
“TAM PO DD H

402 801°3
356 740°8
314 917°4
277 038°7
242 826'9

3151°42
2912°28
2 684°74
2 468°34
2 262°66
2 067°84
1 883°84
1 710°58
1 547°85
1 395°67
I 252°99
I TIQ'IT
993°289
875°997
764° 368
6617582
568°103
483°498
407°664
340° 414
281°065
228-961
183°597
144°893

41
H™,

6 PER-CENT.

Commutation Table—(continued).

27 057°71
24 745°43
22 060°69
19 592°36
17 329°69
15 261°86
13 378'02
11 667°44
10 L19°59

8 723°926

7 470°936
6 351827
5 358°538
4 483°441
3 719°074

3 957°492
2 489°389
2005°891
1 598°227
1257°813

212 017°8
184 360°!
159 614°6
(37 553°9
117 961°6

100 631°9

85 370°02
71 992°00
60 32.4°56
50 204°97
41 481°04
34 O10°TI
27 658°28
22 290°74
17 816°30

14.097°23
11 039°74

112°346

85°77 4
O4' 501 5
47°936 6
35°097 4
25°4272
18°084 9
I2°410 5
8°123 1
50184
2°792 3
I°3102
"4529
"079 2

976°748 1
747°787 5
564°190 7
419°298 2
306951 8

221°2344
156°7329
108°796 2
73°698 9
48°271 6
30°186 7
17°7762
9°6531
4°6347
1°842 4
°5322

"079 2
"000 O

8 550347
6 544°455
4.946°228
3 688°415
2711°667
1 963°879
I 399°688

g80°390 I

673°438 3
452°204 0
2905°4711
186'6749
112°9760

64°704 4
34°517 7
16°74] 5
7°088 4
2°453 8
"O11 4!
"079 2
‘000 0

118-026
92°3799
‘711007
53°966 5
40°473 2
29'9166
21°9178
15°786 2
10°973 0
7°27606
4°5588
2°571 6
1°222 5
"4276
. °075 5

R, x
29 014°36 1 55
26 879°54 | 6
24. 835°75 7
22 882°57 8
21 O19°5I 9
19 245°95 | 60
17 561°63 I
15 966° 38 2
14. 459°99 3
13 042°17 4
11. 712°48 =| 65
10 469°86 6
9312°781 | 7
8 239°250 | 8
7 246°9904 | 9
6 333°213 | 70
5 495°648 | 1
4732°298 | 2
4041477 | 3
3421549 | 4
2 870°678 |75
2 386°195 | 6
1 963°687 | 7
1 598°732 | 8
1 286°587 | 9
1 022°279 | 80
‘801"109 4] 1
618°660 5} 2
470°672 7 3
352°646 4) 4
260°266 5| 85
189°'165 9] 6
135°199 3] 7
94°726 2! 8
64°809 6| 9
42°891 8} go |.
27°05 §| 1
16°1325| 2
88559] 3
42971) 4
1725 5] 95
"5030] 6
"075 5| 7

G

10 |4°788 107/1°976 677/5°211 893) 0°023 325 |6'020 0051975 173
"022 925 15°995 1781975 061
"022 633
022 458
"022 390

=~

wv

aN aN & So re)
h&wWYeM OO OW GHr~T AN PS © DW OO woOr~ An PH HHO OO HT AN BW HK O WO GBT An SO DW

wn

Log D,

"764 784
741 859
"719 226
"696 768

674 378
"651 940
‘629 339
“606 460
"583 184

"559 493
"535 549
"511 426
"487 255
°$63 118

"439 935
"414.957
"390 855
"366 656
"342 341

"317 912
"293 355
268 714
"243 990
"219 188

"194290
"169 273
"144 110
"118 791
"093 333

A
(Log ep,)

977 975
"977 367
O77 542
"977 O10

977 562
"977 399
‘O77 I21
"976 724
"976 309

976.053
"975 850
"975 829
"975 863
975917

"975.922
‘975 898
975 801
"975 685
975 571

975 443
975 359
975 276
"975 198
975 102

‘974 983
"974 837
"974 681
974 542
‘974.410

"974 312
974 232
974 124! |
973 95114"
"973 763] °

"973 483
"973 154
"972 819
"972 494
"972 150
"971 827
‘O71 SII
‘O71 121
‘979 657
"970 154

42
H™

§ PER-CENT.

Logarithms and Co-logarithms of D,, N,, and M,; with

their Differences.

Colog D

"235 216
258 141
"280 774
"303 232
"325 622
°348 o60
°370 661

"393 540
"416 816

"440 507
"404.454
"488 574
"512745
"536 882

"560 965
"585 043
"609 145
033 344
657 959
682 088
"706 645
"731 286
"756010
-780 812

"805 710
"830 727
"855 890
"B81 209
"906 667

"932 257
"957945
"983 713

009 589
035638

061 875
088 392
"r15 238
"142 419
"169 925

"197 775
"225 948
"254 437
283 316

°312 659

A
* (Colog tp,)

"022 438
"022 601
022 879
023 276) °
"023 OQI

"023 947
"024120
"024 171
"024 137
"024 083
"024 078
"024 102
"024 199
"O24 315
"024 429
"O24 557
"024 O41
"024 724
"024 802
024 898

"O25 O17
"025 163
025 319
025 458
"025 590] °
"025 688
025 768
"025 876
"026 049
026 237

026 517
"026 846) °
"027 181
"027 506) °
"027 850

028 173
028 489] °
028 879| -

"029 343
"029 846] °

> Go Go Y © ~ mt
0 OO ONTO SPO HHO OH CONIA FO HHO OO GH~3 Ann PW YH O

Ym O © wrr atn OP

43

H™,
65 PER-CENT.
Logarithms and Oe Dit of D,, N,, and M,; with

ir Differences.

"Log M,; A Colog M;

024 827 |3'934 706 | 985 293 |4°065 294

a!
ban

014 707

Ln

~

&>

&o rN)
> pope d OO COrN3I AN PW NH OO OH GOAN HhO HHO YH CON AM HO WB & O

tn
ko bBo WO Cn Qn

"235 111
"261-509
288 051
"314 742
"341 595
°368 626
"395 847
"42.3277
"450 906
"478 760
°506 841
"535 159
"563 726

°592 555
°621 662

"651 063
681 772
"710 805
"741177
"771 908
"803 018
834 533
"866 483
"898 go2
931 817
"965 263
“999 268
5°033 862

"069 076

"104 O47

"141 516
178 831
"216 946
"255.918
"295 805

024 939
025075

025 231

025 402].

025 582
025 762
025 940
026 104
026 254

026 398
026 542
026 691
026 853
027 031

O27 221
027 424
027635
027 854
028 081

028 318
028 567
028 829
029 107
029 401

029 709
030033
030 372
03973!
O3L TIO

031 515
031.950
032 419
032915
033 446
034.005
034 594
035 214
035 871
036 569
037 315
O38 115
038 972
039 887

040 866

"919 999
"908 300
"898 839
"890 791
883 422
876 021
"867 930
"858 559
"847 323
"834 192
"820 057
B05 365
"79° 734
776 596

763 044
"749 841
736 860
“723 791
"710 558
"697 184
683 637
670 096
656 585
643 126

629 668
"616 143
"602 466
"588 624
"574 690

"560 703
546 782
"532 998
"519 273
°505 437
"491 463
"477 125
"462 320
"447 065
"431416

"475 359
"398 976
"382 307
"365 221
347 587

988 301
99° 539
991952
992 631

992 599
991 909
990 629
988 764
986 869
985 865
985 308
985 369
985 862
986 448

986 797
987 o19
986 931
986 767
986 626

986 453
986 459
986 489
986 541
986 542

986 475
986 323
986 158
986 066
986 013

986 079
986 216

986 275
986 164
986 026

985 662
985 195
984 745
984 351
983 943
983 617
983 331
982 O14
982 366
981775

"080 oo!
"O91 700
“IOI 161
"109 209
"116 578
"123 979
"132 070
“141 441
"152677
"165 808
"179 943
"194 635
"209 266
"223 404

236.956
"250 159
"263 140
"276 209
"289 442
"302 816
"316 363
"329 904
"343 415
"350 874

"370 332
"383 857
"397 534
"$11 376
"425 310

"439 297
"4.53 218
"467 002
"480 727
"494 563

"508 537
"522 875
"537 680
"552 935
"568 584
"584 641
"601 024
617 693
"634779
652 413

o11 699
009 461
008 048

007 369

007 401
008 og!
009 371
O11 236
013 131

014135
014 692
014631
O14 138
O13 552
O13 203
012 981
O13 069
013 233
013 374

013 547
O13 541
O13 511
053 459
013 458

O13 525
013 677
013 842
013 934
013 987
013 921
013 784
013 725
013 836
013 974
014 338

org 805

O15 255
015 649
016057
016 383
016669
017 086
017 634
018 225

bq

wo

& )
WO O~X An PH YH CO WO CN? Qn FW Ym CO WOO DW Qi Hw WO ww O

&o

>

AwWH AO YO ON QAM LW DH & O

44

H™,
5 PERCENT.

Logarithms and Corlogarithims of D,, Nz, and M,; with
their Differences.

x | LogD, Colog D, | Log N,; A -| @

A A
(Logtpz) (Colog tp,)

a a ee en,

ve) CO CO ~I ~ oN
0 OO wer ANN RON HO CO CHO"! Qn POH HO CO CH? Qt BPH HD mH

Ve)
~Sratn — © D HH

"820 584
754427
"684 394
"610 302
"532 007
"448 807
"359 701
"263 865
"161 046
"05° 559
1°933 069
"B09 570
"680 667
"545 275
“05 299
"257 316

"968 948) -

"968 265

"967 537
"966 683

965 726
"964 669
"963 502
"962 216
"960 896

959 527
*958 100
"956 585
955 054
95.3 166

"95° 924
"9.48 204

"944 98013002 924
"94.1 246
937 282

"933 843
"929 967
"925 908
"921 705
"916 800

"910954
‘QO4g 104
897 181
"889 513
"882 510
876 501/2°066 931
"871 097
"864 608
"860 024
"852 017

"836 474
"815 933
"790 845]1°090 277
"745 389
"071 393] °554 043
538 673] *882 650
"242 85810°343 977

I°rO1l 11g

372 926

"403 978
"435 713
468 176

"501 493
"535 767
"571 098
"607 596
"645 380

"684 484
"724957
"796 857
"810272
"855218

"902 052
"951128

"057 944
"116 698

"179 416
"245 573
"315 606
*389 698
"467 993

"551 193
"640 239
"736 135
"838 954
"949 441

"190 430
"319 333
"454 725
"594 791

"742 684,
"906 210

"299 432

"031 052
"031735
"032 463
"033 317

"034 274
"035 331
036 498
"037 784
"O39 104
"040 473
‘O41 Oo
"O43 415
"O44 946
"046 834 |!3°940 7121932 663
"O49 076
"051 796
"05,5 020
058 754
"062 718
"066 157
"079 033
"074 092
"078 295
"083 200
“089 046
095 896
"102 819
"110 487
"117 490

"123 409
"128 903
"135 392
"139 976 |1°867 461/816 231
"147 983
"163 526
"184 067
"209 155
"254 O11
"328 607

55 3°657 495]1'969 579.4342 5°5 07030 421 14°663 3291958 091
"627 074
"596 022
564 287
"531 824

"498 507
"464 233
"428 go2
"392 404
"354 620

"315 516
"275 043
"233 143
"189 728
"144 782
"097 948
"048 872
2°997 076
"942 056
*883 302

"621 4201956 974
"578 3941955 783
"534 1771054 503
"488 680/953 136

"441 8561951 679
"393 495/950 124
"343 0191048 468
292 0871946 704
238 7911944 816

"183 6071942 785
"126 3921940 584
"066 9761938 187
"005 1631935 549

"873 3751929 524
"B02 899'926 147

"729 046/922 566
"651 6121918 823
"57° 4351914 930

485 365/910 728
"396 093/906 214
"302 307/901 331
203 6381895 978
"099 616/890 167

2°989 7831883 995
873 778877 648
"751 426)87 1 097
"622 523/864 547
"487 070/857 783
"344 853)850 307

"195 160/841 454
"036 6141830 847

683 6921796 124

"$79 8161770 023
"249 8391734 827
0°984 666/681 351
"666 017/599 364
"265 381|460 653

"461 327 11°726 034/172 847
"757 142 |2°898 881

BG bb

© CO (oe) ~r “w On
0 OO OT Qn BOY aA QO DO GON AWM HPO HH OC WHO Wr Qn

‘oO
“STAM OO HD

45

H™,
5 PER-CENT.
Logarithms and Co-logarithms of D,, Nz, and M,; with
their Differences.

041 909 |3°329 362

018926) 55
6 043 026] °310 436/980 307| 689 564| 019693} 6
] 044 217| °290 7431979 483| °709257|020517| 7
8 045 497 | °270 226/978 621} °729 774] 021 379 | 8
9 046 864| °248 847/977 578] °751153|022422! 9

Lo,
oO
vn
@)

048 321 | 2264251976 403
049 876| *202 828/975 108
O57 532] "177 936) 973 687
053 296} °151 623/972 128
055 184] °123 7511970 585

057 215] ‘094 336] 969 028
059416] 063 364/967 451| °936 636
061 813} *030 815/965 809| "969 185
064 451 |2°996 624 | 964 218 |3°003 376
067 337| ‘960 842 | 962 176] 039 158

070476] °923 018/959 706| °076 982
073 853| ‘882 724|956 641
077 434| “839 365 |952976| °
081 177] "792 341/948 709] °
085.070] °741 050/944 229

089 272| *685 279] 940 556
093 786| 625 835/936 405
098 669| *562 240/932 117
104 022] °494 3571927753
109 833) “422 110/922 615

"773 575
"797 172
"822 064
848 377
"876 249

"905 664

023 597
024 892
026 313
027 872
029 415

030972
032 549
034191
035 782
037 824

040 204
043 359
047 024
O51 201
055771
059 444
063 595
067 883
072 247
077 385

bh O& HN

“_ fw DN

=,
Le)

© Or An PW WN HO WO COON AN

~
wT!

~~
Oo Or Qn POH WO OO CO~N A

“I

80 116.005] °344 7251916 416| ° 083 584 | 80
I 122 352] °261 1411909 085 OgogIs| |
2 128 903 | °170 226/901 753 098 247} 2
3 135 453| °971 979 | 893 598|_ 106 402] 3
4 142 217|1°965 577 | 886 297 113 703| 4

85 149 693 | °851 874 | 880 250] - 119750] 85
6 158 546) °732 124/875 043 124957| 6
7 169 153} °607 167 | 868 746} ° I31254| 7
8 183 769| °475 913 | 864 883 135117) 8
9 203 876| °340 796 | 857 483 142 517| 9

go 229977 | "198 279 | 842 048 157952 | 9°
I 265 173] °040 327 | 821 603 178 397| I!
2 318 649 |0°861 930] 796 917 203083} 2
3 400 636) °658 847/751 351 248649] 3
4 539 347| *410 198 322947} 4
95 827 153 | °087 251) 543 768) *g12 749 | 456 232 | 95
6 1°631 O19 | 246 672 |0°368 O81 | 753 328} 6
7 2°877 691 7

46

H™.
5 PER-CENT.
Values of Annuities, and Single and Annual Premiums for
Assurance of a Unit.

z az A; ay Az Wz
10 | 17°056 8 | °140 152]° 10°135 2|°469 751 | ‘042 186
I |16°997 8 | °142 960] "482 350] °044 372
2 | 16°919 2|°146 704] ° "495 132] °046 701
3 1 16°824 3) °151 221° "508 088 | -o49 185
4 |16°717 2/°156 323 |° 521 223) °051 841
15 |16°6o1 7|°161 824 |° 60 "534 464 | “05.4670
6 | 16°482 0] 167 525 |° I "547 766-057 678
7 |16°362 4|°173 217] ° 2 "561 092 | ‘060 875
8 | 16°247 5/°178 690 |° 3 574 406 | 064 269
9 | 16°142 0| 183 713 1° 4 °587 667 | °067 868
20 | 16°047 1 |°188 234] ° 65 "600 924] °071 704
I |15°9568]°192 536° 6 614.215] °075 815
2 115°868 0| -196 761 |° 7 "627 583 | "080 246
3 115°7762|°201 131)" 8 | 6°537 81 °641 055 | 085 045
4 | 15°677 9 | "205 816): 9 | 6'2507 |°654 727 | ogo 298
25 115°571 8| 210 867 |° 70 | 5°962 5|°668 453 | "096 008
6 115°459 5|°216 214|° 1 | 5°675 8|°682 105 |-102 176
7 115341 7|°221 822 ]° 2 | 5°3947|°695 488 | -108 759
8 |15°2208|*227 580)° 3) 5°1234|°708 411] °115 690
9 |15°097 3|°233 462 ]° 4] 4°865 6|°720 688 | -122 868
30 | 14°970 9 | "239 482)" 75. | 4°621 5|°732 310] °130270
1 1 14°841 8/°245 630]° 6 | 4°38191°743 717|°138 188
2 |14°708 2|°251 989 | ° 7 | 4°14871°754 824] °146 604
3 | 14°569 8) 258 580): 8 | 3°920 5|°765 693 | °155 614
4 |14°426 1/265 423 |° 9 | 3°6959)°776 431] "165 376
35 |14°277 3|°272 508)" Bo | 3°475 2 |°786 897 | -175 836
6 | 14°123.9|°279 814] I | 3°2660|°796 857 | 186 792
7 113°966 4|°287 313 |° 2 | 3°073 0|°806 048 | ‘197 gor
8 | 13°8049|°295 007 |° 3 | 2°893 9|°814 578] -209 196
9 | 13°638 3 | °302 940 |° 4 | 2°7322)|°822 277/220 320
40 |13°4660/°311 142]° 85 | 2°581 0/829 4781 231 635
1 | 13°286 5|°319 688 |- 6 | 2°4299 |°836671 | °243 934
2 113°0098 7 | °328 632 |° 7 | 2°269 6|°844 305 |°258 230
3 | 12°902 9 |°337 957 |" 8 | 2°099 8 | 852 389 | °274.979
4 | 12°700 5|°347 596|° 9 | 1°898 4|°861 980 | -297 396
45 |12°491 4|°357 551 |" 9° | 1°669 2 | 872 897 | °327 030
6 | 12°297 9 |°367 720]° I | 31°432 4|°884 173 |°363 506
7 | 12°060 8 |°378 058) ° 2 | 1°188 3 |°895 793 | "409 347
8 | 11°839 8 | 388 583]: 3 923 5 |°908 404 | °472 260
9 | 11°613 9|°399 339° 4 | °659 8|'920961 | "554 856
50 | 11°383 1)°410 331] ° 95 | °4061)°933 041 | 663 544
I {11°45 9 |°421 622] ° 6 "174.9 |°944.051 | °803 498
2 | Lo'Qor 6 | "433 256)" 7 952 381 |°952 381
3 | 10°651 2 °445 182)"
4

10°395 7 ,°457 347°

H™.

SIX PER-CENT.

11 933 387
II 109 282

10 337 597

> Go &o iS) bo bm
0 OO OAM POH 4 OC WO DWT Qn #O Hw CO OO CHr~!3r At SW DH = OC

b&&® DN

>

HOODY HO OO DOr Qn

23.1377
21 683°1
20 320°1
19 o41°8
17 839°9
16 709°4
15 6464
14 646°8
13 708°3
12 827°6
12 001°2
11 225°6.
10 497°3
9 812°89
9 169°84
8 566°18

7.999°83.

7 469°23
6972°55
6 507°27
6 070°63
5 660°82
5 275°29
4912°29
4 579°73
4 249°75
3 948'18
3 665°28
3 400°17
3 151°42
2917°74

43 ©
H™

@ PERCENT.
Commutation Table.

824 105°6
771 685°0
722 428°9
676 115°!
632 550°2
593 594°7
553 O10°2
516 756°2
482 687°0
450 700°3

420 697°5
392 572°
366 217°2
341 §24°2
318 386°5

296 703°4
276 383°3
257 345
239 501°6
222 792°2

207 145°8
192 499°0
178 790°7
165 963°2
153 962°0

142 730°4
132 239°!
122 426'2
113 2564
104 690°2

9 615 168
8 939 53
8 306 503
7714938
7161928
6645 172
6 162 485

5711 784
5 291 087
4 898 515
4 532 298
4190774

3 872 387
3 575 O84
3 299 300
3 041 959
2 802 457

2 579 665
2 372 519
2 186020
2 OOI 230
1 835 266

1 681 304
I 538 568
I 406 329
1 283 903
1170646

96 690°3 5
89 221°12
82 248°57
75 741°30
69 670°68
64 009°86
58 734°57
53 822°28
49 251°55
45001°81
41 053°63
37 388°35
33 988°17
30 836°76
27 919°02

1 065 956

969 265°8
880 04.4°7
797 799°2
722 054°9
652 384°2
588 374°3
529 639°8
475 817°5
426 565°9
381 564°1
340 510°5
303 122°!
269 134°0
238 297°2

6031°27
5 773°14
5 575°84
5 421°59
5 294°21
5 180°72
5 069°71
495161
4 818°83
4 664'81

4 491°45
4 312°31
4 133°87
3 963°71
3 806°13
3 66121
3 525°59
3 397°43
3 273°40
3 152°70
3 935°53
2921°53
2 812°13
2 707°32

2 607°06

2 510°78
2 417°86
2 327°66
2 240°05
2155°44

2073°97
1996°19
I 922°30
1851'°6y
1 783°38
1 717°20
I 652°09
1 587°69
I 524°19
1 461°92
I 400°90
I 341°49
1 283°85
1 227°56
I 172'26

154 662°1
148 630°8
142 857°7
137 281°9
131 860°3
126 566°0
121 385°3
116 315°6

III 364°0

106 545°2
10l 8804
97 388°9

93 07660

88 942°73

84 979°02
81 172°89
77 51168
73 986°09
7° 588°66
67 315°26
64 162°55

61 127°03

58 205°49
55 393°36
52 686°04
50 078°98
47 568°20
45 150°34
42 822°68

40 582°63

38 427°18
36 353°21
34 357°02
32 434°72
30 583°03

28 799°65
27 082°45
25 430°36
23 842°68
22 318°49
20 856°56
19 455°66
18 14°17
16 830°32
15 602°76

~~ hs

>]

> ~ Go Go w
hRONe CO OO CON An PW DH OO OO Dr AW PW KH | CO OO Dr AM BO KP HO WO C~1 Qtr BW DW HO

CA

49

H*™,
6 PER-CENT.
Commutation Table—(continued).

z D, N- Se M,; R; x
55 |2 69826 |25220°76 2103782 |1117°94 114430°50 | 55
612 491°99 )§=(|22 72877) |185157°4 1106440 113 312°56 6
71229815 |20430°62 |162428°7 |ro1r61 = |12 248°16 7
8j2116°05 {18 314°57 |141998'0 | O59°601 [11 236'55 8
9|194512 [1636945 [123 683°5 | 908'445 |10276'95 | 9
6o |r 784°48 [14 584°97 [107 314°0 857°911 | 9 368°502 | 6o
111633°51 (12951°46 | 9272905 | 807°950 | 8 S10°5Q1 I
2|149r°68 [11.459°78 | 79777°59 | 758°577 | 7 702°642 | 2
31135850 [10 101°28 68 317°8r | 709°835 | 6944°065 | 3
4.11233°56 | 8867718 | 5821654 | 661°788 | 6334°230 | 4
65 |t 16°71 775V'0Ir | 49 348°82 | 614°761 5 572°442 65
6 |1007°774 | 6743°268 | 41 597°81 | 569°007 | 4957°681 | 6
7| 906'428 | 5836840 | 34854°54 | 524°734 | 4388675 | 7
8 | 812°460 | 5024°380 | 29017°70 | 482°073 | 3863941 | 8
9 725°669 4298°710 | 23993°32 | 441°270 | 3381°868 | 9g
70 | 645°339 | 3653°371 | 19694°61 | 402°016 | 29407598 | 70
I} 579°947 | 3082°424 | 16041°24 | 364°153 | 2538582 | 1
2] 501°976 | 2 580°448 | 12958°82 | 327°499 | 2174°429 | 2
3] 438°073 | 2142°375 | 1037837 | 2927010 | 1846931 | 3
4| 379°°33 | 1763°343 | 8235'992) 257°766 | 1554°920 | 4
73| 324968 | 1438°374 | 6472;649] 225°157 | 12977155 | 75
6| 276419 | 1161'°955 | 5034275] I95°002 | 1071'998 | 6
7| 233°034 | 928922 | 3872°320] 167°263 | 876°9964) 7
8} 194°630 134/201 2 943°398] 142°050 709°733 8] 8
g| 160°990 573°302 2209°107] I19°'426 567°684 I} 9
80 | 131°669 441°633 1635°805| 99°2175| 448°2579) 80
1| 106°248 335°386 | 1194°172| 812495] 3490405] 1
2] 84°3930] 250°993 858°786| 65°4089) 267°7910] 2
3] 65°973 185°019 607°794| 51°7666| 202°3821) 3
4| 506720) 134°347 422°775| 40°1992) 150°615 5) 4
85{ 382967) 96°0503) 288°428/ 30°692 1) 110°416 3) 85
6} 285460 67°504 3 192°378] 23°1092 79°7242| 6
7{ 210148 46°489 4 124°874| 17°1938 566150) 7
8 15°241 1 31°248 3 78°384| 12°609 6 39°421 1} 8
9} 10°9377 20°3106 47°136, 911689) 26°8115! 9
go 77059 12°604 7 26°825 6°556 3 17°642 6] go
I 5°238 2 7°366 5 14°220 4°5247 11°086 3] 1
2| 3°3962 5°970 3 6854; 2°979 3 6°561 6} 2
3 2°078 4 1°8919 2°884 1°853 7 3°582 3] 3
+ 1°45 5 “746 4 "992 1°038 4 1°728 6] 4
95 "5324 "213.9 "246, = 4902 "690 2) 95
6 182 3 70316 032 "1702 "20001 6
7 031 "000 0 "000 "029 8 70298) 7

H

50

H™,
6 PERCENT

_ Logarithms and Co-logarithms of D., N,, and M,; with
their Differences.

Log N, A

om

10 14.746 941/1'972 5015 5°253 059)0°027 439 |5°915 9831971 457

pw LP

Lm

wo

&o

&>
YO CO~?1 (AG APD DH = CO CO Crm Arr HhWwWHKW KF OC W/O CHOr~ Aun

aN
-~ OH Bw = O

aN

POW CO CO CON QAM

"972 958
"973 259 *
"973 427) °
"973 493} °

973 445] °
"973 282) °
"973 005) °
"972 608] -
"972 192] °

971936} °
"971 764) °
O71 712) °
"971 746) °
971 Bol] °
971 895] °
"971 781) °
971 686) -
"971 568) °
971 454) °

"971 327) °
"971 242) °
°971 160) °
°971 081] °

"970 985) °
"970 867] °

"970 720 *

"970 425] °
"970 294] °
"970 195] °
"970 116) °
"970 007
"969 834] °
"969 647) °

"969 366} °
"969 038} °
"968 702] °
"968 377) °
"968 033] °

*967 712) °
"967 393] °
"967 005] °
"966 540) °
"966 038} °

"028 195] °
7028 219] °
7028 314] °
"028 432
°028°546) °

028 673
028 758
7028 840] °
028 gI9| °
"029 O15

"887 440,971 355
"026 750] °
"026 573
026 507

"026 555
"026 718
"026 995
"027 392
"027 808

"028 064) °
"028 236] °
‘028 288
028 254] °
028 199

0 CO~I OTR P&H HO

w
AO Wm O

ww

&o
PO we O OO CON AN

&s
$s WO conT Qui

A puown

ABO Bw O OO CONT A

Colog N,

10 |6°084 017

&o iS) Cs) ~
OO cor~ TF Qi RPO HW HO HO C~? Arr BOHN HO LO DWN Qh PO YD

Go

"112 560
"141205
"169 979
"198 905

"227998
"257 207
286 714
"316 334
*346 112

°376030
"406 081
"436 261
"466 578
"497 945

"527 077
"558 488
"589 490
"620 692
"652 100

683 724
715 571
"747 955
"779 988
812 586

"B45 405
878 640
"912 126
945 938
"980 094

40 5014 617

KR Ow

ey
koObH O © CON An

"049 532
"084 872
"120 667
"156.950
"193 753
"231 106
269 038
°307 580
"346 770
386 648
"427 264
"468 672
"510931
8.54 100

A

51

H™,
6 PER-CENT.

their Differences.

Log M;

028 543 |3°780 408

028 645
028 774
028 926
029 093

029 269
029 447
029 620
02977

029 918
030051
030 180
030 317
030 467
030 632
030 811
031 002
031 202
031 408
031 624

031 847
032 084
032 333
032 598
032 879

033 175
033 486
033 812
034 156
034 523

034915
035 340
035795
036 283
036 803

037 353
037932
038 542
039 190

039 878).

040 616
041 408
O42 259
043 169
044 142

"761 412
*746 311
"734 127
423 801

"714 390
"704 983
694747
"682 942
668 834

"652 387
"634 710
"616 357
598 102
"580 484

"563 625
"547 231
"531151
"515 000
"498 683
"482 234
"405 611
449 036

"432 539

“416151

"399 809

"383 432
"366 920
"350258
"333 537
°316 803
"300 203
*283 821
"267 568
"251 244
"234 820
°218 033
"200 765
"183 039
"164.925
"146 408
"127 589
"108 415
“089 042
"069 024

A

Colog M,

981 004 |4°219 592

984 899
987 816
989 674
990 589

99° 593
989 754
988 195
985 892
983 553

982 323
981 647
981745
982 382
983 141
983 606
983 920
983 849
983 683
983 551

983 377
983 425
983 503
983 612
983 658

983 623
983 488
983 338
983 279
983 266
983 400
983 618
983 747
983 676
983 576
983 213
982 732
982 274

981 886 |-

981 483
981 181
980 826
980 627
979 982
979 394

238 588
"253 689
"265 873
"276 199
285 610
"295 017
"305 253
"317 058
*331 166

"347 613
"365 290
"383 643
“401 898
"419 516

"436 375
"452 769
"408 849
"485 000
"501 317
‘317 766
"534 389
"55° 904
"567 461
"583 849
“600 191
616 568
633 080

"649 742
666 463

°683 197
"699 797
"716179
"732 432
"748 756
"765 180
"781 967
"799 235
816 961
"835 075
"B53 592
872 411
"B91 585
"910958
"930976

Logarithms and Co-logaritims of D,, Nx, and M,; with

A

018 996

O15 101
o12 184
O10 326

009 411
009 407
O10 236
o11 805
014 108

016 447

017 677
018 353
018 255
017 618
016 859

016 394
016 080

O16 151
016317
016 449
016 623

016575
016 497
016 388

016 342

0163977
O16 512
016 662
016721

016 734

016 600
016 382
016253
016 324
0:6 424

016 787
017 268
017 726
018114
018 517
018 819
O19 174
O19 373
020 018
020 606

Io

= > © © w S) mt
0 O GIA bBWHYHO © DNIAMN $0 08 HO © ONT OAN PWD A

&FWwWvHm OO CH Qn PO WN mw

’ 52

H™,
6 PER-CENT.
Logarithms and Co-logarithms of D,, Nz, and M,; with

5

SB pews

~“T

oo ~~?
BON mH CO OO C™ At BPW YD = O WO CONT A

oO
iO com Qn

5 3
6
7
8
9

Lam
e e e e

their Differences.

Colog D,

"603 454
638 622
‘074 474
"711 054
*748 487
"786 877
"B26 325
866 940
"908 840

"964 832
*361 378) °

952 O61
"996 650
"952 4069/3042 667
‘ogo 198
"139 261

"190 212
"243 404

"929 727] °
925 851) -
921 791] °
917 583) °
"912 684) °
"906 837] °
"B99 988} °
‘893 064/2°073 693
"885 397] °
"878 393] °
872 384) -
"866 981} -
"860 491] °
"855 908} °
"847 909) °
"832 3597/1113 175
S11 817] 280 818
786 728] 469 oo:
"741 272) °682 273
667 278) *941 oo!
"534 556/0°273 723

238 7411 °739 167
"500 426

A
(Colog vpz)

"43.1 08411°965 462/4°568 916] 0°034 538 |4'4or 7581954 818
"396 546 8

7035 16

"035 852
036 580
"037 433

"038 390
"039 448
"040 615
"04.1 goo
043 221

"044 589
"046 017
"047 531
"049 063
"050951

"053 192
055.913
"059 136
‘062 871
"066 835

"070 273
"074 149
078 209
"082 412
"083 316

"093 163
"100 012
"106 936
"114603
*121 607

"127 616
133 O19
"139 509
"144 092
"152 100
"167 643
"188 183
°213 272
258 728
"332 722
"465 444
"761 259

Log Nz 4

"356 5761953 706
"370 2821952 515
"262 7971951 237
"214 0341949 872

"163 9061948 4.13
"112 3191946 857
059 1761945 200
"004 3761043 436
3°947 8121941 546
"889 3581939 512
828 8701937 308
"766 1781934 905
"701 0831932 255
"633 3381929 356
"562 6941926 198
*488 8921922 803
"411 6951919 201
"330 896015 441
"246 3371911 535
"157 8721907 317
"065 189!902 790
2°967 9791897 889
"B65 868/892 515
758 3831886 679
"645 062/880 483
"525 5451874 116

"399 661/867 555
"267 216/861 012

"128 2281854 271

1°982 499/846 832
829 331/838 023
"667 354/827 472
"494 826/812 897
°307 723/792 810
"100 5331766 728

0°867 261/731 558
*598 8191678 072

_*276 891/596 056

1°872 9471457 287

"330 234/169 340
"499 574

AP MH

Ve} co co “I ~r A
0 OO ONAN POD FH CO OO OHr~ QAM POH HO [OO CDOr~? An

Ve)
“r CaA(n - GO b Mm

&

on
© Cnr An

A nv
fo)

© cox Qh PO DP

“wr

¢

~~
OO OAT ANn POH mw GO OO G7 Qn fw HW HO

© co ie ¢)
Oo

~~ ae AO ND

Logarithms and Co-logarithms of Dz, N,, and M,; with

Colog N,

5°598 242
"043 424
"689 718
°737 203
785 966
"836 094
887 681
"940 824
"995 624

4°052 188
"110 642
"171130
"233 822
"298 917
306 662

"437 306
“511 108
"588 305
"669 104
"753 663
"842 128
934 811
3°032 O21
"134 132
"241 617

354.938
"474.455
"600 339
"732 784
"871772
2°O17 Sol
°170 669
"332 646
"595 174
"692 277
_*899 467
1°132 739
"401 181
"723 109
0°127 053
"669 766
I°500 426

A

53

- pw
6 PERCENT.

their Differences.

Log Mz

045 182 |3°048 418
046 294| °027 104
047 485| *005 014
048 763 |2°982 091

050128

051 587] °

053 143

054 800! °
056 564| °
058 454] °
060 488} °

062 692
065 095
067 745

070 644| °

073 802
077 197

080 799 | °

084 559
088 465
092 683
097 210
102 11
107 485
113 321

IIQ 517
125 884
132 445
138 988
145 729
153 168
161977
172 528
187 103
207 190

233 272
268 442
321 928
403 944
542 713

I

Oo

"958 299

"465 398
"411226

"352 485
"290 038
"223 399
"152 440
077 100

"996 588
‘go9 821
"815 637
"714.050
"604 217

“487 027
"363 785
"235 373
"100 703
"962 317
"816657
"655 592
"4/74 110
"268 030
"016 376

830 660 |T'690 377

2

*230996
"474 268

A

Colog M,

978 686 14-951 582

977910
977 °77

"972 896
"994 986

976 208 |3°017 gog

975 143

973 942
972 616
971 158
969 561
967 987
966 411
964 822
963 174
g6I 592
959 538

957 O4f
953 926
950 188
945 828
941 259

937 553
933 361
929 041
924 660
919 488
913 233
905 816
898 413
890 167
882 810
876758
871 588
865 330
861 614
854 340
838 935
818 518

793 920
748 346
674001

"O41 701

066 558
"092 616
"120 000
"148 842
"179 281

"211 204
"244 883
"280 O61
°316 887
"355 295

"395757
°438 716
"484 790
534 602
"588 774

"O47 515
"709 962
776 Oo:
"847 560
"922 goo

2°003 412

“Ogo 179
"184 363
"285950
"395 783
°512973
"636215
"764 627
"899 297

1'037 683
*183 343

"344 408
"525 890
‘731.970
"983 624

540 619 |0°309 623

243 272

"79 004

1°525 732

A

O21 314
022 ogo
022 923
023 792
024.857
026 058
027 384
028 842
030 439
032013

033 589
035178
036 826
038 408
040 462

042 959
046 074
o49 812
054.172
058 741
062 447
066 639
070 959
075 340
080 512

086 767
094 184
1or 587
109 833
117 190
123 242
128 412
134 670
138 386
145 660

161 065
181 482
206 080
251654
325 999

459 381
756 728

Qn
Oo O~ An

~y ~ n fo
© Cx~7 Qin © LP & O O ONT OAN Bw HHO

1° <)
Oo

BD
Oo ONT Ah SG DS

\O
sak GW WF O

54 ~

H®™,
_6 PERCENT.
Values of Annuities, and Single and Annual Premiums for

Assurance of a Unit.

Lome

|

Ss

<*)

&o

oO
I
2
3
4
5
6
7
8
9
fe)
I
2
3
4
25
6
7
8
9
)
I
2
3
4
5
6
7
8
9

aN
r@)

Pio db = O © coxr oth AO Pb

ir

11°307 5
Im’r33 9
10°9567
10°75 4
10°589 3
10°398 1
10°200 7

9°996 0

9°785 0

9°568 7

"108 o1!
"IIO 131
"113 201
"117 062
"121 525
"126 404
"131495
"136 581
“I41 443
"145 836

‘149 701
"153 325
"156 854
"160 520
"164 499
"168 851
"173 502
"178 420
183 488
"188 678
"194 008
"199 406
"205 14!
"211055
217 233
223 665
"230 332
"23.7 204

"244 285
"2.51 622

*259 252
267 256
"275 695
"284 557
"293 772

"303 348
"313.175
"323 207
"333 407
"3.44 003
"354 823
*306 000
377497
389 526
"401770

006 854
°007 005
"007 226
"007 505
"007 830
“008 190
008 570
"008 954
"009 325
"009 664

"009 965

"010 250
"OIO 530
"010 823
"OIl 145

"O11 499
“ors 883
"O12 292
"012 720
"O13 164

"013 625
"O14 104
"014 608
"O15 142
"O15 709
"016 308
"016 939
"017 602
018 297
"019 032

‘org 811
"020 645
O21 545
"022 513
"023 546
024 647
"025 810
‘027 032
028 319
"029 683

"031 130
"032 677
"034 330
"036 117
"038 015

ir
OO cnr Qh

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SO pownn sd

OO coO~r Qn && HHO OO O™N OA

Oo ©
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pov nS

wr A

$14 318
"427 128
"440 186
"453 487
*467 038
"480 761
"$94 608
"508 539
"522 513
"536 487

"550 512
"564 635
"578 903
"593 350
608 087

622953
"637 804
652 420
666 578
680 063

‘692 857
"795 456
"717 762
"729 844
"741 825

753 54°
"764 719
"775 952
"784 655
"793 322
801 431
"809 542
818 177

"827 344
"838 286

"850 809
"863 794
"877 225
"891 873
"906 519
"920 661
"933 605
"943 396

"040 042
042 203
"044 508
"046 969
"049 602

"052 409
"055 396
7058 571
"061 O41
"065 515
"069 326
"073 411
"077 816
"082 591
"087 826

"093 520
"099 676
"106 247
"113 163
"120 318
"127 687
"135 571
"143 949
"152919
"162 641
"173 063
"183 975
"195 026
*206 247
"217 273

"228 454
"240 595
"254 708
°27 I 237
"293 421
°322 800
"358971
"404 435
466 886
"548 889
"656 794
795 725
"943 396

a sspuuarssssenyoeteaspeennunnaanennn east
SS

ONE «LIFE.

ee
a ne ee a—eiarey

II.

HY.

HEALTHY FEMALE LIVES.

but in|

SS)

&> r)
WO C~? Air BW HY tO ©) Cr~T Ar BPW HHO |} CO~ Qn PO HD HO

bo

way
re) © Grr acr powennd

O&O WH

lt;

100 000

HF.

Elementary Values.

Log ?,

5°'000 000
4°998 634

996 801

"994 563
“992 OO!

989 196
986 216
"983 143
"980 062

"977 056

"974 138
"O71 141
"967 964
"964 509
"960 661

"956 365
"951 643
"946 673
‘941 511
"936 262

"931 O41
"925 864
"920 749
"915 695
"910 603
"905 483
"900 345
"895 146
"889 884
"884 557

*B832 809
"826 684
*820 490
"814 168
"807 623
"800 786
"793 502

Log pz

1°998 634
998 167
"997 762
"997 438
"997 195

‘997 020
"996 927
* "996 919
"996 994
"997 082

"997 003
996 823
"996 545
996 152
995 7904

"995 278
"995 030
"994 838
994.751
994.779

"994 823
"994 885
"994 946
"994 908
"994 880

"994 862
"994 801
"994 738
"994 673
"994 608

"994 516
"994 428
"994 344
"994 252
"994 149

"994 945
"993 982
"993 928
"993 875
"993 806

"993 678
"993 455
"993 163
"992 816
"992 382

A

999 533

999 595
999 676
999 757
999 825

999 997

999 992
000 075

000 088

999 921

999 820
999 722
999 607
999 552
999 574

999 752
999 808
999 913

000 028
000 044

000 062
000 O61

999 962
999 972
999 982

999 939
999 937
999 935
999 935
999 908

999 912
999 916
999 908
999 897
999 896

999 937
999 946
999 947
999 931
999 872

999 777
999 708
999 653
999 566
999 611

Colog 2,

5°000 000

"00! 366
"003 199
"005 437
"007 999
"O10 804
"013 784

"016 857
"019 938
"022 9.44
"025 862
7028 859

"032 036
"035.491

"039 339

"043 635
"048 357
053 327
7058 489
"063 738

"068 959
"074 136
"079251
"084. 305
"089 397

094 517
"099 655
"104 854
"TIO 116
"115 443

*120 835
"126 319
"131 891
"137 547
"143 295

"149 146
"155 101
"161 119
°167 IQ!
"173 316

"179 510

"185 832
“192 377
"199 214
*206 398

> 4X) & w wo = =
0 OO OY Qtr HOw WMO WO HY AM BW HWM CO OO DO~Y? Ann HO DN HO

- & WD

aN

unr
hOoObHR OO OO CN AN

band

bat

we

O>

O>

fe)
I
2
3
4
5
6
7
8
9
fe)
I
a
3
4
25
6
7
8
9
°
I
2
3
4
5
6
7
8
9

7 pPuPRnsd

tr
AOH He CO WO CO~ Qn

0°00! 366
"oor 833
"002 238
"002 562
"002 805

"002 980
"003 073
"003 O81
"003 006
"002 918

"002 997
"003 177
°003 455
"003 848
"004 296
"004 722
"604.970
"005 162
"005 249
"005 221

"005177
"005 115
"005 954
"005 092
"005 120
7005 138
"005 199
7005 262
"005 327
"005 392

"005 484
"005 572
7005 656
"005 748
7005 851

"005 955

"006 018
"006 072
7006 125
"006 194

"006 322
"006 545
"006 837
"007 184
7007 618

000 467
000 405
000 324
000 243
000 175

000 093
000 008
999 925
999 912
000 079

000 180
000 278
000 393
000 448
000 426

000 248
000 192
000 087

999 972 |

999 956

999 938

999 939
000 038

000 028
000 018

000 061
000 063
000 065
000 065

000 092 |.

000 088
000 084
000 092
000 103
000 104

000 063
000 054
000 053

000 069

000 128

000 223
000 292
000 347
000 434
000 389

314 | 2°496 930

420

"623 249
797 57°
"764176
“800 717
"824 126
"834 421
"832 509
818 885
"802 774

‘B11 575
"833 784
"866 678
‘QI0 O91
"953 760

"99° 339

3°007 748

‘O19 116
"021 189
"013 680

‘004 751

2°904 317

"984.077
982 271

"979 548

975 891
975 891
"975 891
975 891
975 89x
977 724
"979 093
"980 003
"981 366
"983 175
"984.977
°983 626
981 366
"979 993
"977 724

"980 458
"989 005

3°OOI 30!

"015 779

‘ "033 826

126 319
084 321
056 606
036 541
023 409
O10 295
998 088
986 376
983 889
008 801

022 209
032 894
043 413
043 669
036 579
O17 409
o11 368
003 073
992 491
991 O71

989 566
989 760
998 194
997 277
996 343

000 COCO
000 000
000 000
000 C0O
oo! 833

oor 369

000 910
oor 363
oor 809
oo1 802

998 649
997 740
997 127
998 631
002 734

008 547
O12 296

014.478
o18 047
013 838

873 681
915 679
943 394
963 459
976 591
989 705
OOT O12
O13 624
O16 111
991 199
977 791
967 106
956 587
956 331
963 421
982 591
988 632
997 927
007 509
008 929

O10 434
O10 240

oo1 806
002 723
003 657
©00 000
000 000
000 000
000 000

998 167

998 631
999 O90
998 637
998 191
998 198
OOI 351
002 260
002 273

Oo! 369
997 266

991 453
987 704
985 522
981 953
986 162

> wo ©o i) 4 , bm bm
O OO @ TF AW PW HYHO WOO CHAN PON HO WH CNA Po WwW tO

KR pw en

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PhOoObH NM O OO ON A

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5
6
7
8
9
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I
2
3
4
75
6
7
8
9
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I
2
3
4

fo)
© oak pwpn d © c~7F Qn

4°785 984
777977
"709 O14
*760 890
“754 787
"742 214
"731 806 |
"720 424
"707 885
"694 043
"678 791
"662 248
"644 439
*625 168
"604 269

"581 540
“556 676
"528 904
"497 773
"402 937
"423 999
*380 718
"333 004
282 509
"229 426

"174 234
"117 835
060 547
3°999 218
"931 051
"853 881
"764 699
°661 150
"543 696
"413970
"275 772
"136 086
‘000 000
2°885 361
"799 341

"7IO 117
608 526
"480 007
"301 030
"000 000

HF,

Log pz

1°99! 993
"991 637
"991 276
"990 897
"GQO 427
"989 592
‘988 618
"987 461
986 158
"984 748

"983 457
"982 191
"980 729
‘979 TOI
"977 271
975 136
"972 228
968 869
"965 164
961 062

959719
"952 286
"949 595
"946 917
"944 808

"943 or
"042 712
"938 671
"931 833
"922 830
‘910 818
"896 451
"882 546
870 274
*861 802
"860 314
"863 914
"885 361
"913 980
"910 776
"898 409
871 481
821 023
"698 970

Elementary Values—(continued).

A

999 644

999 639
999 6215
999 530
999 165

999 026
998 843
998 €97
g98 590
998 709

998 734
998 538
998 372
998 170
997 865

997 092
996 641
996 295
995 898
995 957

995 567
997 219
997 412
997 891
998 793

ggg iii
995959
993 162
99° 997
987 988

985 633
986 09 5
987 728
991 528
998 512
003 600
021 447
028 619
996 796
987 633
973 072

949 542
877 947

Colog f,

5214 016

"222 023
"230 386
"239 110
248 213

"257 786
"268 194
"279 576
"292 115
"305 957

“321 20y
"337 752
"355 5601
"374 832

"395 731
°418 460
"443 324

"471 096

"502 227

"537 063
°576 001
619 282

666 996
"717 491

"77° 574

"B25 766

"882 165
"939 453

4°000 782

"068 949
"146 119
"235 301
"338 850
"4.56 304
“586 030
"72.4 228
"863 914

3°000 000

"114 639
"200 659
“289 883
"391 474
"349 993

_*698 970

2°000 000

i)

\O foo] co “TY ~T
WO CO~7 An #&$WwW Ne OO OO CO~T Awn HOD 4 CO Oo ad pwnn SD

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tn
Oo Or An

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Oo pwnd

1 CN? Qin P&H we CO OO COVA

© 0] oe)
3S O CO™IF Qtr POH BD —& O

ve)
OO DW~3 Qui SLO HD

Colog pz

0°008 007

"008 363
"008 724
"009 103

"999 573

"010 408
"o11 382
"O12 530
"013 842
"O15 252

"016 543
017 Bog
"O19 371
"020 899
"022 729

"024 864
"027 772
"O31 131
"034 836
7038938
043 281
"047 714
"050 495
053 083
055 192
"056 399
"057 288
"O61 329
"068 167
"077 170
089 182
"103 549
“117 454
"129 726
"138 198
"139 686
"136 086
"114,639
"086 020
"089 224
"IOI 5QI
"128 519

178977
°301 030

HF.

Elementary Values—(continued).

000 356
000 361
000 379
000 470
000 835

000 974
OOT 157

COI 303
OO! 410
OO! 291

001 266
oor 462
Oo! 628
001 830
002'135

002 908
003 359
003 795
004 102
004 343

004 433
002 781
002 588
002 109
OO! 207

000 889
004 041
006 838
009 003
012 O12

014 367
013 905
012272
008 472
oor 488

996 400
978 553
971 381
093 204
012 367

026 928

050 458
122053

Log d,

3°047 664
"058 426
"068 186
"077 731
“ogo 258
"516 608
"144 574
°174 641
"204 391
"231979
"251 395
"266 232
"281 942
"297 104
“311754

"326.950
"348 694
"368 845
"384 712
"396 199
“401 056
"397 940
"373 404

"343 499
*306 211

"2.59 833
"209 783
°180 126
"161 368.
"142 702

"122 544
"OOK 315
"035 830
2'955 688
"B49 419

‘715 167
“366 848
"365 488
"139 879
"068 186

"029 384
‘O17 033
-008 600
*9000 000
*000 O00

A

O10 762
009 760
009 545
O12 527
026 350

027 966
030 067
029 75°
027 588
Q19 416

04 837
O15 710
O15 162
014650
015 196

O21 744
020 151

o1 5 867
o11 487
004 857

996 884
975 524
969 945

962 802

953 622

949 950
97° 343
981 242
981 334
979 842

968 771
044 515
919 858
893 73!
865 748

850 681
799 540
774398
928 307
961 198

987 649
991 567
99! 400
ooo 000

—A

989 238
990 240
99° 455
987 473
973 650

972 034
959 933
970° 250
972 412
980 584

985 163
984 290
984 838
985 350
984 804
978 256
979 849
984 135
988 513
995 143
003 116
024 476
039 045
037198
046 378
050050
029 657
018 758
018 666
020158

031 229
055 485
080 142
106 269
134252
149 319
200 360
225 609
071 693
038 802
O12 351
008 4.33
008 600
000 000

“J ~T WN A
© COr~3 An OH HO /O CON AK POD

Ve) roe) foe)
0 orxr ae wwe CO DW CON Qw b&w WB HO

60

Probabilities of Lavt over, ond of Dying in, a Year, and the
Average ig eae of Life, at each Age.
Vx 2 Gx
| Pe |(mp)| * |? pom | (1—ps)
TO | °996 860 | °003 140 55 | 981 732| °018 268
I |°995 787 | (004.213 6 | "980926 | 019074
2 | "994 862 | °005 138 7 |°980113 | *o19 887
3 | °994.117 | °005-883 8 1°979258| *020 742
4 | °993 562 | 006.438 9 |°978 199] °021 801
15 | 993 162 | °006 838 60 | ‘976 319] °023 681
6 | °992 950 | °007 050 1 |°974132]| °025 868
7 |°992 931 | °007 069 2 |°971 541 | °028 459
8 | °993 100 | ‘006 goo 3 | °968 631 | °031 369
9 | "993 305 | 006.695 4 |°965 491 | °034 509
20 | °993 122 | ‘006 878 65 | °962 623 | °037 377
I | °992 711 | °007 289 6 |°959 822 | °040178
2 | "992 080 | °007 920 7 1°9560599| °043 401
3 | 991 178 | :008 822 8 |°953.018] 046982
4 | "990 158 | cog 842 9 | "949 010] “050990
25 | 989 186 | -o10 814 7° |°944357| °055 643
6 | 988 621 | ‘orr 379 I |°938053 | °061 947
7 | °988 185 | -orr 815 2 |°930826| *069174
8 | 987 986 | -o12 014 3 |°922921| °077079
9 | "988 049 | 011 951 4 1°914245| °085 755
30 | °988 150 | ors 850 75 |°995146| 094 854
1 | '988 293 | ‘or1 707 6 | "895.955 | °104045
2 | ‘988 430 | ‘orr 570 7 | °890236| *109 764
3 | °988 343 | ‘or 657 8 | °884946| 115054
4 | °988 280 | -or1 720 9 | 880660] ‘119 340
35 | ‘988 240 | -or1 760 80 | °878 2141 °121 786
6 | -988 t00 | ‘orr goo 1 |°876420] ‘123 580
7 | °987 957 | ‘012 043 2 | 868 302] °131 698
8 | "987 810 | ‘o12 190 3 |°854739| *145 261
9 | °987 659 | ‘012 341 4 | °837 201 | *362.799
40 | °987 453 | ‘012 547 85 | °814 363 | °185 637
I | °987 253 | ‘012 747 6 | °787 864| °212 136
2 | 987 061 | ‘012 939 7 |°763 038 | *236 962
3 | °986 850 | ‘013 150 8 1 °741779| °258 221
4 | °986 620 | -or3 380 9 (°727448 | °272 552
45 | °986 382 | -or3 618 go | °724.960] 275040
6 | ‘986 237 | 013 763 I 1°730994]| ‘269 006
7 | °986 117 | -o1r3 883 2 |°768 000] ‘232 000
8 | "985 995 | ‘014 005 3 | °820 312) *179 688
9 | 985 841 | 014 159 4 | °814.286| °185 714
50 | 985 547 | 014 453 95 |°791 423 | °208 577
I | "985 043 | 014.957 6 | °743 843 | 256157
2 | "984 380 | °o15 620 7 | 662252) °337 748
3 | °983 594 | 016 406 8 | °500000] *500 000
4 | ‘982 613 | 017 387 Q | ‘000 000 | 1°000 000

HF.

THREE PER-CENT.

= bus
© CO~ An HPO HW HO

Ny

Go

°
I
2
3
4
5
6
7
8
9
re)
I
2
3
4
35
6
7
8
9

wl > >
© Cnr An —-& WB & O

& OO DH = O

Dz

74 409°4

72 0153
69 623°2
67 2480
64.905°2
62 609°!
60 369°9
58 198°4
56 103°8
54 °93'9
52 166°8
59 299°°0
48 478'1
46 693°2
44 953°
43 195°!
41 483°5
39 817°0
38 200°5
36 642°3

35 149°9
3372107
32 356°3
31 050°4
29 7946
28 587°8
27 428°7
26 312°9
25 238°9
24205°0
23 210°0
22 251°3
21 327°8
20 438°7
19 582°4

18 757°7
17 963°3
17 200°!
16 467°3
15 763'8
15 087°9
14 436°7
13 806°6
13 195°!
12 600°6

Nz

1729035"
1657 020°
1 587 397°
I 520 149°
1455 244°

I 392 634°
I 332 265°
I 274 006°
I 217 962°
1 163 868°

I 1II 702°
1 061 403°
1012925°

62
HF

3 PER-CENT.
Commutation Table.

Ss

26613 982°
25221 348°
23 889 083°
22615 017°
21 397 955°
20 233 186°

19 127 484°
18 060 082°

34 562 827°
32 833 792°
31176 772°
29 589 375°
28 069 226°

956 2314
921 298'2

878 103°!
836 619°6
796 802°6
758 602°!
7219598
686 809'9
653 088-2
620 731'9
589 681°6
559 887°0

531 299°2
503 870°5
477 557°6
452 318°7
428 113°7

404 903°7
382 652°4
361 324°6
340 886'0
321 303°5

302 545°8
284 582°5
267 382'4
250915"!
235 1513
220 0634
205 626°7
IQI 820°!
378 624°9
166 024°3

17 047 157°
16 080 926°
15 159 628°
14 281 524°
13 444 905°
12 648 102°
11 889 500°

II 167 540°
10 480 731°
9 827 642°
9 206 910°
8 617 229°

8057 342°
7 526 043°
7022 172°
6 544 615"
6 092 296°
5 664 182°
5 259 279°
4 876 626°
4 515 302°
4174 416°

3 853 112°
3 559 506°
3 205 984°
2.998 6o1°
2 747 686°

2 512 535°
2 292 472°
2 086 845°
1 895 025°
1736 400°

21 8819
21 655'0
21 360°5
21013°2
20 629°1

20 223°4
19 807°7
19 394°5
18 995"!
18 6§9°3
18 267'°7
17 919°4
17 563°4
17 390°5
16 790°6
16 361°2
159°7°7
15 449°4
14 992°7
14 547°!

14 121°9
13717°5
13 334°3
12.970°8
12 619°4

12 280°4
11 954°0
11 637°!
IT 320°4
Il 030°9

10 740°7
10 458'0
10 182°6
“9 914°65
9 653°72
9 399°33
9 151°33
8 gII°29
8 679°45
8 455°55
8 238°85
8 027°13
7 817°49
7 608'11
7 397°94

M,; | R, |

744 2328
722 350°9
700 695'9
679 335°4
658 322°3

637 693°2
617 4C€9'8
597 662°!
578 267°5
559 272°4

540 653°2
522 385°5
504 466°1
486 902°7
469 712°2

452 921°6
436 560°4
420 652°7
405 203°3
390 210°7
375 663°6
361 5§41'°6
347 8241
334 489°8
323 519°
308 899°6
296 619°3
284.665°3
273 028°2
261 698'8
250 668°1
239 927°4
229 469'5
219 286'9
209 372°3

199 718°5
190 319°2
181 167°9
172 256°6
163 577°!
155 121'6
146 882°7
138 855°6
131 038°1
123 430°0

-. Oo &> w w =~ ma
So" CNT QAM POH HO OO CON An #O YH HO OO DW An PW BW HO

Ww Ww = O

pf

tn
mow bP tO WO CON An

x D,

55 |12 020°9
6 11 457°6
7 |1OQII'7
8 |10 383°2
9 | 9 871°67

937521

nN
Oo

~ ~ oN
wm #&WH DD ow

© O~ 73 Ain OO Be O OO CON A

feo) (oe)
OO con~ Qn BO WD = CO

‘e
© wr~r A OH YW ww O

8 886-60
8 404°58
7927°57
1455°23
6 988°31
6 53117
6 086°18
5 652°45
5 229°99
4 818'75
4 418°08
4.023°68
3 63626
3 258°23
2 892°06
2 5+1°49
2 210°74
1910°76
I 641°67

1.403°64

1 196°79

1018°34
858°476
712°§00

579°951
4.57°823
_ 350°196
2 59°430
186°83 5

131°954

92°8750
65°913 6

49°147 3
391419

30°944 3
23°7707
17'171O
T1040 3

5-359 4

63
HF,

3 PER-CENT.

Commutation Table.

Nz

154.003°4

142 545°9

131634°2

121 251°0

III 379°3

102 004'1

93 117°50
84 712°92
76 785°35
69 330°12
62 341'°82
55 810°65
49 72447
44 072°02
38 842°03
34 023°28
29 605°20
25 581°52
21945°26
18 687°03

15 79497
13 253°48

889 561°8
787 557°7
694 440°2
609 727°3
532 942°0
463 611°8
401 270°0
345 459°4
295 7349
251 662°9
212 820°9
178 797°6
149 192°4
123 610°9
101 665°6

82 978°57
67 183°61
53930713

42 887°40
33 755°42
26 265°10
20 178°43
15 288'55
11 417°02
8 403°964
6 103°307
4 381°700
3.117°916
2 204°329
1550971
1 082°848
747°478
504°984

Mz

7 185°24

5 972°04
5 759°86
6 549°18
6 340°09
6I31I5
5915760
5 692°42
5 460°20
5 218°76

4. 968°98
4715°39
4 460°62
4 204°17
3 940°34

3 687°43
3427°11
3 161°39
2 891°17
2 619'05

2 347°78
2 081°44
1 824°7!
1 589°12
1 375°69
1185°48
1 O19'51
875°920
745°713
624°641
§12°041
407°679
313°387
232°820
167°781

118°342

83°1069
58°85 7

R;

116032°1
108 846°8
101 874°8
95 114°93
88 565°75
82 225°66
76.094'51
7017891
64 486°49
59 026°29
53 807°53
48 838°55
44 123°16
39 962°54
35 458°37
31 512°04
27 824°61
24 397°5°
21236°11
18 344°94
15 725°89
1337811
11 296°67

9 471955
7 882°832

6 507°145
5 321°668
4 302°156
3 426°236
2 680°523
2 055°882
1 543841
1136°161

822°774

589°954

422°173
303°830
220°723

328°403
200°969
112°677
55°33
21°759
5°359

“000

44°004 I
35°430 2
28°3727
22°1064
16°193 2
10°562 7

5°203 3

161°873
117°869
82°438
54°'066
31°959
15°766
5°203

~] A fo, i
© mak pwn 8 © wr ot | 8

~“r

Oo ON ON SO He EO HO GW Qh PO Ww HO

8

8

9

© wr Ae. GO vw mt O

64
HF, .
S$ PERCENT.
Logarithms and hee Die of D,, N,, and M,; with
. ences.

Colog D,

A
(Colog ep-)

10 (4°871 628|/1°985 797|5°128 372/0°014 203

I | "857 425] ‘985 329] *142 575] ‘014671
2 | °842 7 5¢ "157 246) ‘O15 074
3 | °827 680) - "172 320] “O15 400
4 | ‘812 280] ° "187 720] ‘015 642
15 | *796 638] ° "203 362] ‘o15 817
6 | *780 821) “98 "2IQ 179] “O15 QI0/ °
7 | °764911| ° 235 089] 015 918
8 | °748 9931 ° "251007| ‘O15 845
9 | ‘733 148] ° "266 852] “ols 754
20 | ‘717 394! ° 282 606] ‘015 834
I | °701 560) ° "298 440| “o16 015
2 | 685 545) ° "314.455| “016 292
3 | 669 253! ° "330 747

4 | °652 568) - "347 432| ‘O17 133
25 | °635435] ° "364 565) °017 559
6 017 87° , °382 124] *o17 808
7 e 1 e@ °*3 32 ‘Ol7

8 | *582 069] ° Ao O31 018 86
9 | °563 983] ° "436017| ‘018 059
3° | *545 924 "454.076, oI8 014
I | "527910 ° "472 O90) °O17 952
2 | 509958) - 490 042| ‘017 891
3 | “492.067! ° "507 933| “017 929
4 | °474 138] ° "525 862] *or7 958
35 | °456 180) ° *543 820) °017975
6 | *438 205 ° 561 795| 018 036
7 | °420 169) - "579 831

8 | °402 070) ° "597 930] °

9 | °383 906 ° "616.094! °
40 | *365 676) - "634 324] °

I | °347 355] ° "652 645] °

2 | °328 946 - "671 054) °

3 | °310453] ° -689 547] °

4 | ‘291 867] - *708 133] °
45 | °273 £79) ° "726 821) °

6 | °254 387] - "745 613] °

7 | °235 533| ° "764 469] °

8 | +216 622] - °783 378) °

9 | °197 660) - "B02 340] °
50 | °178 629) ° 821 371

I | “159 469) ° "B40 531 °

2 | 140087] ° "859 913] °

3 | ‘120 413] ° "879 587

4 | °100 392] ° "899 608] -

"237 804|981 524
219 328.981 358
"200 686/981 200
“181 soe

"085 634
065 904/980 08

"045 9881979 892
“025 8801979 697
"005 5771979 504
"016 685 |5°985 0811979 319
"964 4001979 146
"943 5461978 982
"922 5281978 823
*QOI 3511978 663
"880 st 499

"105 554580 442

"858 513
"836 837
"814 9721977 932
"792 904.977 71

779 6181977 482
"748 100

"725 3391976 980
"792 319

81 050
ogo!

80 270

78 32
978 135

977 239

976 707

"371 3481971 200

"342 5481970 532
*313 o80\969 814
"282 8941969 048
"2.51 9421968 230
°420 172/967 358

bug be

wo

2

&o )
OO cr~7 An SPO Ne OO HO OW? An BO YD HO OO CNT At Hh WD =O

>
oO

a”
& & bw mH O © cr? Oth hb OD

65

HF.
83 PER-CENT.

Logarithms and Co-logarithems of D,, N,, and M,; with
heir Differences.

2 | Colog N, A

ed

Log M; A Colog M, A

10 |7°762 196

> > wo ww S) iS) al
0 OO O~A AN PWN He CO OO DWN AH BAW WK O OH BW Qn fw VY

POD HO OO Cr~ I An BD DP

in

°780 672

"799 314
"818 114

"837 064
"856 163
"875 409

894 808

"914 366
"934 096

"954 012
"974 120
~ 994 423
6°014 919
035 600

"056 454
"077 4/2
"098 649
"119 986
"141 487
"163 163
"185 028
"207 096
"229 382
"251 900

274 661
297 681
"320 974
"344 555
"308 441

"392 648
"417 196
"442 102
"467 391
"493 085

"519 209
"545 792
"572 867
"600 473
"628 652

"657 452
*686 920
"717 106
"748 058
"779 828

018 476 |4°340 085

018 642
018 800
o18g50
019 099
O19 246
O19 399
O19 558

019 730
019 916

020 108
020 303
020 496
020 681
020 854

021 018
021177
021 337
O21 SOI
021 676

021 865
022 068
022 286
022 518
022 761

023 020
023 293
023 581
023 886
024 207

024 548
024 906
025 289

"335 559
"329 O10
"322 492
"314.479

"305 854
"296 835
‘287
"278 641
269 963
"261 684
"2.53 323
"244 609
"23.5 289
"225 065

19

213 815
"201 607
"188 g12
"175 879
"162 776

"149 894
°137 276
"124 969
"112 967
"IOI 039

089 211
"077 512
"065 844
"054 207
"042 603
"031033

‘O19 447
007 858

025 694 |3°996 278

026 124

026 583
027975
027 606
028 179
028 800

029 468
030 186
030952
031770
032 642

"984 695
‘973 097
"961 484
"949 941
"938 492
"927 142
"915 866
"904 560
893 067
"881 277
869 111

995 474 |5°659 915

994.951
992 882
991 987
991 375

990 981
99° 844
990 962
QQI 322
Qgt 721
991 639
9gI 286
gg0 680
989 776
988 750

987 792
987 305
986 967
986 897
987 118

987 382
987 693
987 998
988 072
988 172
988 301
988 332
988 363
988 306
988 430
988 414
988 411
988 420

664 441

‘670 390
"677 508
685 521

"694 146
703 105
"712 321
"721 359
"739 037

"738 316
"746 077
"755 391
"764711
774.935
-786 185
798 393
"811 088
"824 121
837 224
"850 106
"862 724
875031
"887 033
"898 961
‘910 789
‘922 488
"934 156
"945 793
957 397
"968 967
"980 553
"992 142

988 417 |4°003 722

988 402

988 387
988 457
988 551
988 650
988 724

988 694
988 507
988 210
987 834
987 330

"O15 395
026 903
038 516
"050 059
“o61 508
072 858

"084 134
"095 440
"106 933
"118 723
"130 889

009 320
010 224
OFI 250

O12 208
012 695
O13 033
013 103
o12 882

012618
O12 307
O12 002
o11 928
o11 828

o11 699
or 668
o11 637
oT 604
OII §70

or 586
o11 589
o1r 580
O1I 583
orr 598

O1r 613

OIL 543
OLI 449
O11 350
o11 276

O11 306

O1T 493
Orr 790
O12 166
012 670

EK

hm
©)

rr
Oo WO Om Qn POO HW mw

©

oO OO CONTA BW DY

<S)

&o
OO Orn Ann PO DP

-
re)

aN
Cn hOO HD

ho bKD me CO WO Cr~r A

Log D,;

"059 093 |
037 892|

7016 331!

9 |3°994 391

60

“Tr ~ (©)
OO wor~ An PwOKH WO WO GW Qn —W DW &

0O oo)
pW Dw OW GON An FW Ww O

\O
oO cn Ann

i)

b=

‘971 981.
"948 75

‘924 516
899 140
872 461

844 372
"814991
"784 345
"752 237
"718 501

"682 934
"645 233
"604 624
"560655
*512 982

"461 207
"405 089
"344 537
281 205
"215 285

"147 256
"078 O19
007 894
"933 728
"852724
"762 717
"660 698
"544 311
"414 020
"271457
"120 422
"967 899
818975
"691 499
"502 641

A
(Log ep,)

T 979 156 5° 920 063
978 799)
‘07 8 439|
‘978 060

‘977 59°

979755
6; "975 780
"974 O24
"973 321
‘971911

"970 619
"969 354
"967 892
"966 264
"964 433

"962 299
"959 3591
"956031
952 327
948 225

‘943 882
"939 448
"936 668
"934 080
"931971

"930 763
"929 875
"925 834
"918 996)3°066 272
"147 276

"237 283
"339 302
"455 589
"585 980
"728 543

"879 578

"999 993
"897 981
"884 613
‘$69 709
‘857 437
‘B48 965
"847 477|_
°851 076)2°032 101
872 524
"QOI 142
"897 940
"B85 571

66

HF,
3 PER-CENT.

Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

Colog D,

"940 907
"962 108
"983 669

4°005 60g!

"028 019

"051 264
075 484
"100 860

"127 539
"155 628
"185 009
"215655
"247 763
"281 499
"317 066
"354797
"395 370

"439 345
"487 018

"538 793
"594911
"655 493
"718 795
"784 715
‘B52 744
"921 981
"992 106

"181 025
*308 Sol
"497 359

"509 419
623 848

(Coleg tps)

"029 351
"030 646
"032 108
"033 736
"035 507
"037 791
"040 609
"043 969
"047 073
"051775
7056118
"060 552
"063 332
"06.5 920 )3°960 565
068 029

‘069 237
"O70 125
"074 166
‘o81 004
"OQO 007

"102 O19
"116 387
"130201
"142 563
"151 035

"152 523
"148 924
"127 476
"098 858
"102 060

"114 429
"141 356
"191 814
"313 867 j0°729 115

Log N; aN

0°020 844 /5° 187 5301966 425 55
"021 201
"O21 561
"021 9.40
‘O22 410

"023 245
"024 220
"025 276
"026 679
"028 089

"153 955|905 414
"119 369.964 316
08 3 6851963 120
"046 805/961 813

"008 618]960 413

4°969 0311958 19

"927 9501957 328
"885 2781955 O44
"840 922/953 857

794 7791951 938
"746 7171949 853
"696 5791947 593
"644 1631945 139
"589 3021942 474

"531 7791939 592
"471 3681936 558
"$07 9261933 415
"341 3411930 199
"271 5401926 979

.198 5191923 811
"122 3301920 747
"043 077/917 488
913 935
*874 500909 880

904 918

"784 380
‘689 298/898 585
*587 883;891 124
"479 007|882 845
361 852/874 082

"23.5 9341865 739
"101 673/859 077
2°960 750854 933
"815 683/853 934
"669 617/855 907
"52.5 524/859 178
*384 702,862 242
"246 944/858 349
"105 284/840 636
1945 920812 594
"758 5141707 447
“52.5 961|688 87 5
"214 836/514 279

© Cosr A

N
SO pwnd

oO OO ONT AN

eo) (oe) oe) ~y ~T
0 OO O~ AN PO KHKHO HO GW ANr PHO HW &

ve)
| 0 ow a8 -& b =

67

HF.
3 PER-CENT.

Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

CologN-| A Log M,

013 081

1,

033 575 13°856 441 | 986 919 |4°143 559

55 ,6°812 470

6

7
8

9

\O 0O Oo
OO Om~1 An — GW YH & O

Le)
Oo O~r7 Aw RW DYDD

"846045
"880 631
"916 315
"953 195

"991 382
5°030 969

"072 050
"114.722
"159078
"205 221
"253 283
"303 430
"355 837
*410 698
°468 224
"528 632
"592 074
"658 659
°728 460
"Bor 481
°877 670
"956 923

4°039 435

"125 500

"215 620
"310 702
"$12 117
"520 993
638 148

*764 066
"898 327

3°039 250

"184 317
"330 383
"474 476
615 298
"753 050
"894 716

2°054 080

"241 $86
"474 939
"785 164

1°270 885

034 586
035 684
036 880
038 187

039 587
041 081
042 672
044 356
046 143
048 062
050147
052 407
054 861
057 526
060 408
063 442
066 585
069 801
073021
076 189
079 253
082 512
086 065
Ogo 120

095 082
IOI 415
108 876
117 155
125918

134 261
140 923
145 067
146 066
144 093
140 822
137 758
141 660

159 364
187 406

232 553
311 125

485 721

"843 360
"829 938
"816 187
"802 095

"787 542
"771999
755 297
"737 209
"717 567
"696 267
"673 517
"649 396
623 680
"596 194
"506 724
"534.928
"499 879
‘401 073
“418 144

"370 657
"318 365
"201 195
"201 158
"138 520

073 893
"008 392
"942 464
872 572
"795 931
"799 305
‘610 319
"496 081
"307 O21
224 744

°073 140
"919 037
"709 752
"O43 494
549 374

"4.52 QOL
“344 519
"200 334
"023 774
0°716 278

984.457
983 298
981 g12
980 358
978 700

977 250
975 879
974 284
972 514
979 530
968 204
904951
961 194
957 971
952 513
947 708
942 830
939 993
937 362
935 373

934 499
934072
930 108
923 059
913 674
QOI O14
885 762
870 9.40
857 723
848 396
846 407
850 115
873 742
905 880
993 527
891 618
864 815
814 4.40
692 504

"156 640
"170 062
"183 813
"197 905
"212 458
228 oo!
"244 703
"262 791
"282 433

"303 733
"326 483
"350 O04
"376 320
403 806

"433 276
"465 072
"500 121
538 927
"581 856
629 3.43
681 635
"738 805
"798 842
"861 480
"926 107
°991 608

3°057 536

"127 428
"204 369
"290 695
"389 681
"593 919
‘632979
"775 250
"926 860

2°080 363

"230248
"356 506
"450026

"547 299
655 481
"790 666
"976 226

1283 722

O13 422
013751
O14 092
014 553

O15 543
016 702

018 088
O19 642
027 300

022 750
O24 121
025 716
027 486
029 470

031 796
035 049
038 806
042 929
047 487
052 292
057 170
060 037
062 638
064 627

153 593
149 885
126 258
O94 120
096 473

108 382
135 185
185 560
307 496

YO Cx Qwn

io, \

~I
Oo CMI Ani BW LW WO © ON AD pw en

“I

C
O

ioe)
© Or~ Aw BPW HW w

ve)
& © BH mw

Ve)
eC CONT Aw

wv ome a]
BO be O WO Hr AK HPO WH - O

wo

iS)

> > Go
hWDNeO OO CONTA HW HHO KH Or Qn

tun
hwoOH me O DO CONAN

23°2368
23°009 3
22799 8
22'605 I
22°4210

22°243 3
22°068 4
21°891 8
21°709 I
21°515 7
21°310 5
2I1org
20°894 5
20°693 2
20°503 7
20°328 8
20°167 5
20°01 1 6
19°858 4
19°7029

19°539 5
19°367 0
19°184 3
18991 I
18°791 6

18°584 9
18°3702
18°149 2
17°921 §
17°687 ©

17°445 2
17°1969
16°941 5
16°678 5
16°407 7
16°129 2
15'842 4
15°545 4
15°237 2
14'Q17 2

14°585 4
14°243 3
13°893 3
13°537 2
13°1759

A;

"204 074
*300 700
*306 801
"312473
"317 833
"323 O10
"328 106
"333 249
"338 570
"344 203

"359179
"356 257
"362 296
"368 159
"373 678

"378 774
"383 470
°388 o11
"392 473
"397 003
“401 763
"406 787
"412 108
"417 734
"423 546

"429 567
"435 820
"442 257
448 887
"455719
"462 761
"469 994
‘477 432
"485 093
"492 979
"501 O92
"509 444
"518 095
"527072
"536 392
"540 056
"556021
"566 213
576 586

587 110

68

HF.
3 PER-CENT.
Values of Annuities, and Single and Annual Premiums for

Assurance of a Unit.

Wr

012 133
"O12 524
"O12 89!
"O13 237
"013 57°

013 897
"014223
"014 558
"Ol4 9OQ
"015 287

"015 696
"016.319
"016 547
016.971
"O17 377
"O17 759
018 116
"018 466
018 816
"019 176
"O19 561
"019 973
"020 417
020 896
"O21 400

"021934
022 499
023 095
"023 724
"024 387
025 088
"025 828
"026611
"027 440
"028 320

"029 254
"030 248
"O31 314
"032 461
‘033 699

"035 036
"036 477
"038 018
"039 663
"041 416

z

“A un
Powe OC OO Dr Qn

“A
ww

ba i
© O~r An Pw HHO WO ON A

“lI

ioe]
Oo

© \O CO
OO Cr AN FW HK O OO OQ PO HD

‘| 10°880 2

ay

12°811 3
12°44] 2
12°063 6
11'6776
11°282 7

10°478 4
10°079 4
9°685 9
9°299 5
8:9209
8°545 3
8-170 1
77970
7°.$26 8

"711 042
"721 982
"732911
"743 778
"754 559

"765 225
775 791
"785 697
"795 993
"803 825
"811 801
818 985
"825 387
831672
‘837 982

844 573
*851 870
"860 142
‘S68 648
876 813

884 277
"890 473
"894 889
"897 432
898 021

896 848
"894 826
"892 846
"895 353
"905 174
"916 896
"929 750
"943 056
"959 735
970 874

A; | @,

597 729
"608 509
‘619 507
"030749
"642 251
"653975
"665 677
"677 300
‘688 761
"700 O14

043 278
045 272
"047 422
"049 753
052 289

7055 048
"057 994
‘OO! 132
"064 455
"067 966
"071 671
7075 638
"079 924
‘084 549
"089 543

"004 934
"100 728
"106 785
"113 018
"119 345

"125 637
"131779
*137 678
"143 907
"150 045
"158 268
"167 499
"179 129
"192 O14
"207 311

"222 563
"236 802
"247 975
°254 842
256 485

‘253 235
247 807
"2.42 689
249 201
°278 029

"321352
"385 482
*482 362
"644 076
"97° 874

HF.

THREE AND A HALF PER-CENT.

on!

fw wb HO

7° 891'9
68 279°5
65 692°6
63 145°0
60 650°7

bn

58 222°5
55 869°0
53 599°!
51 420°5
49 338°9

473513
45 435°4
43 578°9
417716
40 003°0
38 269°8
36 575°9
34 936'9
33 356°6
31 84174
39 397°0
29 021°0
27 7114
26 464'5
25 271°5
24 130°7
23 040°5
21.990'5
20 996°7
20 039°4
19 122°8
18 244°3
17 402°6
16 596°6
15 824°5
15 084'8
14 376°2
13 698°9
13 051'°9
12 433°9

11 843°3
11277°4
10 733°1
10 208°2
4] 970114

wo

(

&o
OO GW An Hho HW O HO GW TAH PO HH DO O O™ AN

&

aN
& Bb me O WO Cr An pPwvynd

ws

33 PER-

70
HF

CENT.

Commutation Table. ,

1 499 667

I 431 387
I 365 695
I 302 550
1 241 899
1 183 677
I 127 808
1074 208
1 022 788

973 449°
926 097°9
880 662°5
837 083°6
795 312°0
755 309°0
717 039°!
680 463°3
045 526°4
612 169°8
580 32874

549 930'5
520 910°4
493 199°!
466 734°6
441 463°!

417 332°3
394 291°8
372 295°3
351 2986
331259°2

312 136°5
293 892°2
276 489°6
259 893°0
244 068°5
228 983°8
214 607°6
200 9o8'7
187 856°9
175 423°0

163 579°7
152 302°3
141 569°2
131 361°0
121 659°9

28 199 634
26 699 967
25268 580
23 902 885
22 600 335

21 358 436
20174759
19 046952
17 972 743
16 949955
15 976 506
15 050 408
14 169 746
13 332 662
12 537 350
11 782 O41
1I 065 002
10 384 539

9 739 O12
9 126 843

8 546 514
7996 583
7475 972
6 982 473
6 515 739
6074276
5 656 943
5 262 651
4 890 356
4 539 058
4207 798
3 895 662
3 601 770
3 325 280
3 065 387
2 821 319
2 502 335
2377 727
2176818
1 988 962

1 813 539
1 649 959
1 497 057
I 356 087
I 224 726

M,

17 7812
17 5661
17 288'2
16 962°!
16 603°1

16 225°9
15 841°3
15 460°7
15 094°6
14.7518
14 432°7
14.118°0
13 798°1
13 464° 5
13 108'4
12 728°0
12 3283
11 926°0
Il 527°2
I1 140°0

10772°3

10 424°3

10 096°0
9 786°27
9 488-21
9 202°03
8 927°85
8 662°94
8 406°99
8 159°70
7.920°76
7 688°93
7 404°24
7 246°68
7 03 5°82
6 831°24
6 632°76
6 441°59
6 257°84
6 081°23

5 gi I "r3
5 745°74
5 582°77
5 420°79
5 258'98

R;

563 837°1

546 055°9
528 489'8
511 201°6
494 239°6

477 036°4
461 410°5
445 569°2
430 108°6
415 O14°0
400 262'2
385 829°5
371 7114
357 913°4
344 448'9
331 340°5
318 612°5
306 284°4
294 358'4
282 831°2

271 6O1°2
260 9189
250 494°5
240 398°5
230612°2
221 124°0
211 922°0
202 994°!
104 3312
185 924°2

177 764°5
169 843°7
162 154°8
154 690°6
147 443°9
140 408°1
133 576°8
126 944°1
120 502°5
114.244°6
108 163°4
102 252°3
96 506°54
90 923°76
85 502°97

f. f ss) > i) ) _ =
hH we O WO CONTA PWN HH GO OO ON AN $&O Hm CO OO Dr Atri BO HD | O

wm CO CWO CO~™ AN

tn

3
4

DO pow nm

© «]
O cor~r Qin PW DH

Ve)
© KU AR -&W bd Oo

~7 ~
OO ONFAG B&H HHO 16 CATO

684'602
574339
474309

383°664
301°876
229°794
169°412
20417

85°337 6
59°774 3
42°2170
31°326 3
24°828 3

19°533 7
14°936 6
10°734 8
6°868 7
3°318 2

71
HF

32 PERCENT.
Commutation Table.

112 449°8
103 713°7

95 433°97
87 593°37
80 17504
73 163°82
66 55o°r!
60 325736
54.482°26
49 013°85
43 912°70
39 168°26
34 768'44
3° 7OI"gO
26.957°48

23 524°15
20 391°50
17 552°28
14 998°83
12 721°9g0
10 710°62
8951°676

I 121°375
819°499
589°705
420°292
298'876

213°538
153°704
II1'547
80°220
55°592
35°858
20°922
10°187
3°318
"000

I 103 067°
990 616°8
886 903°!
791 469°!
703 875°8

623 700°7
55° 536°9
483 986°8
423 661°4
369 179°2
320 165°3
276 252°6
237 084"4
202 315°9
171 614°0

144656°5

121 132°4

100 740'9
83 188°62
68 189°79

55 467°89
44 757°28
35 805°60
28 376°56
22 257°19
17 257°62
13 210°86
9 972°568
7 418°880
5 439°532

3 934493
2 813°1t7
1 993°618:
1 403°O14
983°621:
684°740
471°208
317°444
205°898
125°667
70°285
34°427
13°595
3°318'
"000

M,

5 og6°or

4933°46
14 772°46
4 613°37
4 450°24

4 299°98
4 139°57
3 97 4°27
3 803°11
3 626°01

3 443°68
3 259°47
3 975°29
2 890°¢9
2 706°20
2 521°72
2 337°I4
2 149°65
1.959°89
1 769°73
I 581°07
1 396°7 5
I 219°92
1.058°45
g12°859
783°742
671°627
575°°94
487°983
407°375

332°769
263°955
202°082
149°471
107°204

75°2397
52°553 2
37°Ol7 3
27°5542
22°1156
17°660 5
13°7240
10°027 3

6°524 2

3°2060

R,

80 24.3°99

75 147°98
79 214°52
65 442°07
60 828°70

56 372°46
52 072°48
47 932°91
43 958°65
40 155°54

36 529°53
33 085°84
29 826°38
26 751°09
23 860°30
21 154°lO
18 632°38
16 295°23
14 1.45°58
12 185°69

10 415°96
8 834°888
7 438°143
6 218°219
5 159°774

4246°915
3 463°172
2 791°545
2 216°451
1 728°468

1 321094
988°325
724°369
522°288
372°817
265°613
190°382
137°829
100°812

73°258

51°142
33°482
19°758
9°7 39
3°206

in
YO O~r Qu

AN
wpe

Ve) fe °) CO ~? ~T! A
0 OO O31 AN SPW HH CO YO Wr? Qn BW WB HO WO CDWr] CTH

10 cos Oe BO WD

72
HF,
34 PERCENT.

” Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

Log D,

A
(Log tp-)

ey, fe | een OE

10 14°850 597|1°983 693'5° —F "016 307

ee

6°575 9951979 762} 1

oO

"834 290
"817 516;
‘Bop 339)
"782 836

"795 091
‘747 171
"729 158
"7IL 136
"693 189

075 332
657 394
"039 277
*620 882
"602 093
"582 857
"563 194
"543 284
"523 182
"502 992
"482 830
"402 713
| 442 658
| "422 664
"402 631

*382 571
"362 493
"342 353
"322151
"301 884

bud
Po bm

O CONT AM

»
Oo

&o >)
O O~1T AWN PWH HO OO ON DAn FW WH

Go

>
©]

"281 551
"261 12%

as

BwoN sO OO ONAN Fw DW

"178 539
"157 643
"136 684
"115 672
"094 607
"073 473
"052 210
030 725
"008 9471
3°986 823

"983 226

"982 823

"982 497:
"982 255
"982 080
981 987
"981 978
"982 053
"982 143,
"982 o6a
‘981 883|
‘981 605]
"O8r2rr' *3
"980 764
"980 337
"980 ogo
‘979 898
"979 B10
"979 838

979 883!
"979 945
"980 006
"979 967
"979 940

"979 922
"979 860
"979 798
979 733
"979 667
979 57

"979 488.
240 615] °
"220018| °
"199 329) *
"979 104

"979 O4t
978 988)

978.935 |
978 866!

‘978 737

"978 515)

"978 222.
977 876.

"165 710;
"182 484,
‘199 661!
‘217 164

"234 909
"252 829
"270 842
"288 864
°306 81)

°324 668
"342 606
"360 723

379 118

"397 997
"417 143
"436 806
"4.50 716
476 818
"497 008

"517 170
"537 287
557 342
"577 339
"597 369

"617 429
"637 507
"657 947
"677 849
-698 116

"718 449
*738 873!
*759 385
"779 982}
"800 671
"821 46)
"842 357
"863 316
884 328
"995 393
"926 527
"947 790
"989 275
991 053

‘O77 442 47013 177

"016 774!
"O17 177!
"O17 503}
"O17 745
"O17 920
‘018 o13'
"018-022;
"O17 O47
"017 857
"017 938
"018 157
"018 305
"018 789
"O19 236

‘019 663
*OIQ 910
"020 102
"020 190
"020 162

"020 117
"020 055
"O1Q 994
"020 033
"020 060

"020078
"020 140
"020 202)
"020 267
°020 333)

"020 424,
"020 512)
"020 597
"020 689)
"020 790:

"020 896
"020 959
"O21 O12
"O21 065;
"021 134;

‘O21 203.
"O21 485.
"027 778)
"022 124
"022 558)

"155 757979 597
"135 3541979 440
"114 7941979 292
"09-4 0861979 147

"073 233/979 002
"052 235/978 854
"031 0891978 697
"009 786/978 527
5988 3131978 344

"966 6571978 153
"944 810/977 959
"922 7691977 769
"900 5381977 587
"B78 1251977 418

855 5431977
"832 805 977 109
"B09 9141976 958
786 8721976 802
‘763 6741976 635

"740 3091976 454
"716 7631976 259
*693 022/976 048
669 0701975 824
"644 8941975 588

"620 482/97 5 336
"595 818/975 070
"570 888/974 788
"545 67619074 492
*§20 168974 177
"494 3451973 843
*468 188
"441 679
"414795
°387 512

"359 8051971 840
"331 0451071 354
"302 999'970 828
273 8271970 260
24.4 087.969 642
213 7291908 977
"182 706)

"150 969:967 498
°118 467'966 680
08 5 147/905 812

ied ad a keel A i

“262

973 491
973 116
972717
972 293

968 263

(>) ~~
wm POND O WO CON QAM WD =

e

Go

&>
OO O~?3 Qn $e Bw w CO OO COr~T7 A

re)

4

- FO YO =

O COON? Ain

fe)

5

> & YO

73

HF.
31 PERCENT,

Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

Log Mz

020 238 |4°249 961 1 094 715 15°750 039
020 403 | °244676 | 993 073] °755 324

—<——

Colog M,

A

|
0 OO ONT Aw of & Db

wo

i)
Oo Ort An PWD

> So oo
0 ©) ON Aw P&H YD & O

O con at pO Dm

ww
.@ ]

io oO

"033 343
"055 199
‘O77 231
"099 462
"121875

"144457
"167 195
°190 086
"213 128
°236 326

"2.59 OF
"283 237
"306 978
"330930
"355 106

"379 518
*404 182
"429 112
"454 324
"479 832
"505 055
"531 812
5.58 321
"585 205
612 488
640 195
"668 355
697 001
"726 173
755 913
"786 271
817 294
"849 031
881 533
914.853

020 560
020 708
020 853

020 998

5} 021 146

O21 303

021 473
021 656

021 847
022 O41
022 231
022 413
022 582

022 738
022 Bg!
023 042
023 198
023 365

023 546
023 741
023 952

°237 749
"229 479
"220 191

"210 209
"199 789
"189 229
“178 821
"168 845

"159 347
"149 774
"139 818
‘129 189
"117 550
"104 760
“ogo 898
"076 496
"O61 724
"0.46 885
"032 310
°018 047
"004 ISI

024 176 13°990 617

O24 412
024 664
024 930
025 212
025 508
025 823

026 157
026 509
026 884
027 283
027 707

028 160
028 646
029172
029 740
030 358
031 023
031 737
032 502
033 320
034 188

"977 184

963 884
"95° 747
"937 666
"924 O41
"911 674
"808 767
"885 866
872 985
"860 139
847 315
"B34 500
821 605
"808 993
"796 424
"783 991
"771670
"759 346
"7463850
"734 003
°720 go2

991 730°
990712
990 018

989 580
989 440
989 592
990 o24
990 502

990 427
990 044
989 371
988 361
987 210
986 138
985 598
985 228
985 161
985 425
985 737
986 104
986 466

"762 251
770 52
"779 B09
"789 791
"800 211
"810771
"821 179
"831155
"840 653
"850 226
"860 182
*870 811
"882 450

"895 240
"QoQ 102
"923 54,
938 276
953 115

967 690
981953
"995 849

986 567 |4009 383

986 700

986 863
986 919
986975
987 033
987 093

987 099
987 119
987 154
987 176
987 185

987 195
987 298
987 431
987 567
987 679
987 676
987 504
987 213
986 839
986 329

"022 816

7036 116
°049°2 53
"062 334

"075 359
"088 326

"IOI 233
"114 134
"127 015
"139 861
"152 685
"165 500
"178 305
"191 007

"203 576
*216 009

*228 330
240 654
"2535 150
"265937
"279 098

009 976
009 498

009 573
009 956
010 6209

o11 639
O12 790

be =
0 OO G7 An SO HD AO

i)

013 862
O14 402
O14 772
o14 839
O14 575
O14 263
013 896
O13 534

013 433
013 300

013 137
O13 o81
013 025
012 967
012907

eo
09 Ob OY An HOW HW

&2

©“ On h&® Db ww

012 901
o12 881
O12 846
012 824
O12 815

w >
0 OO OX An Pwpnys

|} ROD
1

74

HF.
33 PERCENT.
Logarithms and Co-logarithms of D,, Nz, and M,; with
their Defferences—(continued).

wn
WO ~7 Qn

AN
oO

oe) ~Y ~ Nn
BS wamrn ak pwwnd Oo O37 Ann BOHM O CONAN +O DS

Le)
SI OtA BO BY

Log D,

"941 318
‘918 014
"894 349
"870 306

"B45 793
"820 445
"794 122
"756 643
"737 861

"707 669} °

676 185
"643 435
"609 225

"573 385
"535 716

"495 912
453 199
"407 127
°357 351
°393 473
"245 251
°182 597

"117 162] °
"c49 138] °
2°979 006] *
"907 666] °

835 438

"759 169] °

676 061

"583 951
"$79 829
"361 340
"228 945
"084 279

"931 140

"776 534| °

625 488
"495 999
"394.948

"290 784 | °

"174.252
"030 793

8 10836 876

9

"$20 905

A
(Log tpz)

"976 696
976 335
975.957
975 487

Colog D,

Go!

wl

058 682
"081 986
"105 651
129 694

"154 207
"179 555
"205 878

"233 357
"262 139

"292 331
"333 815
"356 505
"39° 775
"$26 615
"464 284
“504 088
*546 801
592 873
"642 6.49

"696 527
754749
"817 403
‘882 838
"950 862

"020 O94
"092 334
"164 562
"240 831
"323 939
"416 049
"§20 171
638 660
"771955
"915 721
"068 860
"223 486
"374 512
"504 091
"605 052
"709 216
825 748
"969 207

"684 029 |1°563 124

"479 995

A
(Colog epz)

°104.122
"118 489
"132 395
"144 666
°153 139
"154.626
°151 026
"129 579
°100 961
"104 164

"116 532

"143 459
"193 917] °008 044] 512 863

"315 971 |0%520 go5

Log N,;

3°964 265 |1°977 053 |4°035 735 |0°022 947 15°050 950
023 304] “015 836
"023 665 14°979 703
"024 043
"024 513
"025 348
026 323
"027 479
"028 782
"030 192

"031 484
"032 750
"034210
"035 840
"037 669

"039 Bo4
"042 713
"046 072
"049 776
°053 878
058 222
"062 654
"065 435
068 024
"070 132

"O71 340
"072 228
076 269
"083 108
"092 110

942 471
"904 039

864 296

"823 149
"780 500

"736 255

"690 319

"642 590
"592 934
“541 185
"487 165
"430 679

"371 514
"399 449
"244 334
"176058
"104 552
029 814
3°951 904
"870 933
"786 707
698 933

"607 108
"510 316
"407 168
"296 522
"177 548

"O49 751
"O13 549
"779 635
"623 552
“475 490
"329 475
"186 854
"O47 457
"904 285
"743 447

"554 59°
"320 597

A

964877
963 867
962 768
961 568
960 257

958 853
957 351
955755
954 064
952271

95° 344
948 251
945 980
943 514
940 835

937935
934 885
931 724
928 494
925 262
922 090
919 029
915774
O12 226
908 175
9o3 208
806 852
889 3.54
881 026
872 203

863 798
857 086
852917
851938
853 985
857 379
860 603
856 828
839 162
8:1 143

766 007
687 447

GY

On

OO wor~ Amn HPWH HO /H Cr Qn

~

“J

io.)

Loe]
© OI An fw KO

‘Oo
fo)

GO DH

\o

Colog Nz

55 {6949 041

6

"984 164

7 |5°020 207

Spawned ©

~Y
©O ONT An BPO DH OO LO COXA

~

oo
oO

10 cnr A f& YO

7

"057 529
"095 961
"135 704
"176 851
"219 500
"263 745
"309 O81
"357 410
"407 066
"458 815
*512 835
"509 321
628 486

"690 551
"755 666
"823 942
"895 448
970 186

4°048 096
"129 067
"213 293
"301 067

392 892
"489 684
"592 832
"703 478
"822 452

"950 249

3°086 451

"229 365
"376 448
"524 510
"670 525
813 146
"952 543

2°095 715

"256 553
"44.5 410
"679 403
"991 956

8 [1°479 095
9

75
HF

34 PER-CENT.
Logarithms and Co-logarithms of D,, N,, and M,; with

their Dafferences—(continued).

a

Log M,

035 123 |3°707 231

036 133
037 232
038 432
039 743

O41 147
042 649
044 245
045 936
047 729

049 656
051749
054 020
056 486
059 165
062 065
065 115
068 276
O71 506
074 738

077910
080 971
084 226
087 774
og1 825

096 792
103 148
110 646
118974
127 797
136 202
142914
147 083
148 062
146 O15

142 621
139 397
143 172
160 838
188 857

233 993
312 553
487 139

"693 151
678 742
"664 018
648 968

"633 467
616955
"599 257
"580 139
559 429

"537 023
"513 140
"487 886
"461 O17
"432 359
"401 698
"368 686
"332 368
292 232
"247 907

"198 952
"145 117
"086 333
"024 668

2°960 404

"894.173
"827 128
"759 739
‘688 404
"609 994

"522143
"421 531
"395 527
"174 556

"050 211

1°876 395

"720 599
568 405
"440 187
"344 698
"247 004

"137 482
‘oor 183

0°814 530

"505 965

Colog M,

985 920 |4°292 769

984 499
983 488
982 302
980 882
979 290
977 594

976 123
974 740
973 131
971 342
959 339
966 988
963 682
959 864
955 975
951 045
946 165
941 216
938 335
935 736

306 849
"321 258
335 982
"351032

"366 533
"383 045
"400 743
"419 861
"440 571

"462 977
"486 854
"512114
538 983
"567 641
"598 302
"631 314
667 632
"707 768
"752% 993
"B01 048
"854 883
"913 667
"975 332

933 799 |3°039 596

932 955
932 O11
928 665
921 590
O12 149

899 388
883 996
869 029
855 655
846 184

844 204.
847 806
871 782
904 511
go2 306

890 478
863 701
813 347

"105 827
"172 872
"240 2601
"311 5906
"390 006

477 857
“578 409
"694 473
"825 444
"969 789

2°123 605

"279 401
"431 595
"559 813
655 302
"752 996
"862 518
998 817

691 435 [1185 470

"$94 035

A

014 080
O14 409
O14 724
015 050
O15 501

016 512
017 698
o19 118
020710
022 406

023 877
025 260
026 869
028 658
030 661

033 O12
036 318
040 136
044 325
048 955

053 835
058 784
061 665
064 264
066 231

067 O45
067 389
071 335
078 410
087 851

100 612
116 004
13097!
144 345
153 816

155 796
152 194
128 218
095 489
097 594
109 522
136 299
186 653
308 565

Oo) On AN

O pw on

co “I
© On~3 An PW HO WO Dr 3 An BDH HO 1 DOrT A

ve)

OO GOT An PO HN

= bag

v

&

wn > > Go Sd
0 OO ONAN ROW DN & OO Co~r At BPHwOWKW HO OO ONAN $$ YN HO OO DON AN POND & O

AO bb

21°54 3

20°963 6
20°789 2
20°627 9
20°476 2

20°330 2
20°186 7
20°O4I1 5
19°890 7
19°7299
19°558 0
19°382 8
19°208 4
19°039 5
18°881 3

18°736 4
18604 2
18°476 9
18°352 3
18°2256

18°09I 7
17°949 4
17°797 7
17°636 2
17°468 8

17°294 6
17°1130
16°925 2
16°731 1
16°530 4
16°322 8
16°108 7
15°887 8
15°659 4
15°423 5
15°1798
14'928 0
14°666 I
14°393 I
14‘108 5
13°8120
137505 1
13°1Q00
12°868 2
12°5408

250 821]°
"257 268] °
"263 168°
"268 621 |°
273 7591"
278 688 | °
"283 543°
*288 450|°
"293 552)"
"298 990)"
°304 800 |"
*310 728 |"
*316 6231"
"322 335].
"327 685)"

"332 585]°
"3379571"
"341 3591"
345 S751"
"349 8591"

"354 388)"
"359 199 |"
"304 328°
"309 789 |°
375 4507"

"387 341)"
"387 4851"
"393 833]°
"400 396 |
"407 183]°
"414 206°
"421 444)"
"428 O14)"
"436 6371"
"444 617)"

452 857"
"461 372)"
"470 228 |°
"479 460] °
"489 085°
"499 ATT)"
"599 490!"
"520 146]°
"531 025|°
"542 099].

76
HF

34 PER-CENT.
Values of Annutties, and Single and Annual Premiums for

Assurance of a Unit.

fo, tn
BON ee O OO DW~7T Qn

“A

OO oO~ An AO WY BO WO COr~ Qn

~I

Ve) ere) oo
OO WONT AN BW WK O

\O
© GO Ah HP WD ~

12209 4

11°871 8
11°526 3
rr1r718
10°807 7

10°43 5 3
10.062 4
g'691 2
9°324 2
8°963 1
8°608 4
8255 6
7°902 2
7°5499
71994
6°8517
6°509 3
6°182 1
5°8740
5°587 3
5°325 3
5089 2
48791
4°672 §
4°4047

4247 2
4°005 4
3°7392
3°446 3
3°173 1
2°922 8
2°7147
2°506 2
2°480 9
2°461 6
2°502 3
2°5724
2°642 2
2°5608
2'2310

18357
1°400 7
"949 ©
483 1
‘9000 O

As;

"553 306
“364720
"576 406
"588 394
"600 706
"613 300
"62.5 907
638 462
"650 872
"663 084

"675 979
°687 008
"698 958
‘710 873
"722 727
"734 483
"746 060
°757 128
‘767 547
"777 241
786 102
"794 083
"B01 191
808 178
B15 202

"822 559
"830 734
"840 042
"849 642
"858 880

"867 345
"874 383
"879 403
"882 289
"882 942
“881 566
"879 194
"876 833
"879 587
"89° 739
"904 106
918 817
"934 993

949 847
"966 184

"041 887
"043 873
"046 016
7048 341
"050 874

"053 632
"056 579
7059 718
"063 043
"066 554
"070 2.59
"074 226
078 515
"083 144
088 144

"093 545
"099 351
"TO5 419
‘111 660
"117 OQI
"124 280
"130 408
"136.279
"142 474
"149 175
"156 762
"165 966
“177 592
"191 089
"205 812

"221 103
"235 385
"246 592
"253 467
"255.070
"251712
"246 107
"240 742
"247 O19
275 685
°318 828
"382 729
"479 27

"640 451
966 184

HF.

YOUR PER-CENT.

me »
OO co~T Qin P&H | O

e
Cr AO HD & O

ee

&o

> > Go
AHwHKHmO OO OH 7 Ain ABRWDH HO OO CON OA

On
POoOkP HO OO GO Qn

78
HF

4 PERCENT.

Commutation Table.

1 308 776
1244021
1 182 020
1122710
1 066 016

1011 854

g6o 131°2
910 748°3
863 600°4
818 578-7

775 578°4
734 51653
695 321°3
657 932°4
622 298'7

588 372°7
556 104°4
525 430°2
496 284°2
468 596°0

442 290°9
417 297°2
393 546°2
379972°9
349 520°8

329 135°5
309 764'8
291 360°8
273 877°8
257 272°!
241 502°1
226 528°9
212 315°!
198 824°9
186 024°!
173 880°3
162 362'6
151 440°3
141 083'9
13126573
121 958'0
113 138°0
104 784°!

23 182 517
21 873 741
20 629 720
19 447 700
18 324 990

17 258973
16 247 119
15 286 988
14 376 240
13 512 639
12 69.4 061
11 918 482
TI 183 966
10 488 645
9 830712
9 208 414
8 620 od!
8 063 937
7 538 507
7 042 222
6 573 626
6 131 335
5 714.038
5 320 492
4949 519

4 599 998
4270 863
3.961 098
3 669 737
3 395 860
3 138 588
2 897 085
2 670 557
2458 241
2 259 417

2073 392
1 899 512
1737 150
1 585 709
I 444625
I 313 360
I 191 402
1078 264

14 620°6
14 416°6
14 154°3
13 848°0
13 512°5
13 161°5
12 805°4
12 454°8
12 119°!
11 8063

IT 516°5
II 232'2
10044°4
10 645'8
10 328°6

9991°38
9 638°62
9 285°56
8 937°08
8 600°40

8 282°21
7 982°49
770114
7 430°91
7 183°90

ao

431 760°9
417 140°3
402 723'7
388 569°4
374 721°5

361 209°0
348 047°5
335 242°0
322 787°2
310 6681
298 861°8
287 345°3
276 113°)
265 168°7
254 522°9
244 194°3
234 202°9
224 564°3
215 278°7
206 341°7
197 741°3
189 450°1
181 476°6

173 775°4
166 338°5

6 942°1 41159 154°6

6 711°63
6 489°99
6 276°87
6 071°94
5 87490
5 084°64
5 Sorry
5 324°28
5 153°71
4.989'02
4 830°00
4677°57
453077
4 392°31
4258°63
4 129°28
4.00244

973 480°2| 3 876'97
876 603°2| 3 752°24

152212°5
145 500°9
139 O10°9
1327340
126 662°1
120 787°2
II§ 102°5
109 601°4
104.277°1

99 123°41
94 134°40
89 304°40
84 626°82
80 095°05

75 702°74
71 444q°1i
67 31483
63 312°39
59 435°42

iw bh om

mn PFPobwKmMO OO Or QM” APO HD HO

ee

©o

> > &>
PRWwWbwne OO OO ONAN BPO HHO OO CON OQ

oy
hO Ww HO WO CO~? QAM

| 55

79
HF.

4 PER-CENT.

Commutation Table—(continued).

in
\O Crxr Qwa

“2 ~r “Wn AN
ir POH HO OO CT AN P&S HO

oO O~ a

oe)
oO

eo)
WO GT QM &@OO HD

‘oO

ad Dz N, Sz M,; R,
7065°63 182 333°06 |787 204°5 |3627°22 [55 683°18
6 16669°77 175 663°30 |704 871'°5 13 503711 [52 055°96
7 |6290°91 {69 372°39 16292082 13 380°78 148 552°85
8 1592865 163 443°74 1559 835°8 |3260°48 = 145 172°07
9 15 582°39 157 861°35 [496 392°0 13 142°24 [41 OII'59
Go 15 250°66 152 610°69 [438 530°7 |3.025°22 138 769°35
I 1492916 [47 681°53 |385920°0 |2905'°67 135 744°12
2 1461697 143 064°56 |338238'5 [2 783°06 = 132 838-45
3 14313°05 = 138751°51 |2905173°9 | |2650°72 0055°39
4 |4017°07 134 734°44 |256422'4 |2526°63 |27 398°67
65 13 729°27 13100516 |221 688'0 |2 393°33. 124.8 72°04
6 1345181 27 553°35 11906828 [225931 [22 478°70
7 13.185°70 §=|24 367°65 [163 129°5 [2 125°96 |20219'40
8 12 930°23 «121 437°42 [1387618 |1993°01 [18 093°44
9 1268515 3118 752°27 [117 324°4 =|1860°64 =| 16 100°43
7° j2450°23 [16 302°05 | 9857211 11 728'99 =| 14. 239°80
I |2224°89 = |14.077'15 | 82.270°07 |1 597°89 = [£2 S10°81
2 |2006°80 |12.070°36 | 68 192°92 |1 465°37 | 10912°92
3 |8 79613 [1027422 | 56122°56 [1 33189 | 9447°551
4 |1593°93 | 8680°290) 45 848'34 |r198'77 | 8 115°662
75 |§ 40120 | 7279'095| 37 168°05 |1067°34 | 6916°893
O 1 219°51 | 6059°589) 29 888°96 | 939°541 | 5 849°555
7 |1050°60 5 008-991) 23 829°37 | 817°537 | 4Q10°014
8 | 899°308 | 4109°684| 18 820°38 | 706°654 | 4092°477
9 | 765°230 | 3344°454| 14710°69 | 6077165 | 3 385°823
80 | 647°988 | 2696°465| 11 366'24 | 519°355 | 2778°658
I} 547°185 | 2149°281| 8669°773! 443°475 | 2259°302
2 | 461119 | 1688°162| 6520°493| 378°454 | 1815°828
3 | 384991 | 1303°171| 4832°331| 320°062 | 1 437°373
4} 336°410 986°761} 3529°161| 266°288 | 11177312
85 | 254714 732°050] 2 542°400| 216°758 851024
6 | 199°449 532°601] 1 810°350) 171'293 634°265
7 | I5L095 381°506} 1277°749| 130°610 462'°972
8 |} 110°857 270°650| 806°243| 96°183 4) 332°362
9 79°068 44 r91°581 625°593| 68°6588) 236179
99 | 55°3059) 136°275) 434012) 47°9374] 167°520
I 38°552 5] 97°723 297°737| 33°311 1] 119°583
2 27°097 7|' -70°625 200°014| 23°3391 86°271
3 20°010 Of = 50°61 5 12y°389| 17°2943 62°932
4] 15°7836 34°83t) = 78°74) 13°8369) = 45°638
95 12°358 1 22°473 43°943| r'0o184| 31°80r
6] g'4043} 13°069 21°470| 8°5399] 20°783
7 6°726 3 6°342 8°402 6°223 6 12°243
8 4°283 1 2°059 2059} 4°0392 6019
9 2°059 2 “000 ‘000 1"9800 1°980

© ~~ aw mW WH | O

$0

HY.
4 PERCENT.
Logarithms and Co-logarithms of D,, Nz, and M,; with

their Dufferences.

Io

I

2

3

4

15 979 987] °
6 "979 804) °
7 "979 885) °
8 "979 960) °
9 "980 O50] °
20 "979 969) °
I "979 790] °
2 979 512) °
3 "979 118) °
4 "978 671) °
25 "978 245] °
6 "977 996} °
7 "977 805] °
8 977 718) °
9 977 7451 °
30 "977 789) °
I ‘977 852) °
2 977 913] °
3 "977 875} °
4 "977 846] °
35 ‘977 829) °
6 "977 768] °
7 "977 704! °
8 ‘977 640] °
9 ‘977 5741 °
40 "977 483) °
I 977 395] °
2 977 311] °
3 "977 218] °
4 "977 116] °
45 "977 O12} °
6

7

8

9

50

I

2

3

4 "975 349] °

"018 868
"019 270| °
"Oly 596] °
"org 838

"020 O13

"020 I15
"020 040] °
"019 950] °
"020 O31
"020 210
"020 488 | °
"020 882
"O21 329] °

"021755
"022 004] °
7] *022 (95
"022 282
"022 255

"022 211
"022 148
"022 087
.022 125
"022 154

"022 171
"022 232
"022 296] °
"022 360
"022 426

"022 517
"022 605
"022 680] °
"022 782
"022 884] -

"022 988
"023 052
023 105
"023 159
"023 226

°023 357
7023 578
"023 870] °

"149 477
"118 1501968 060

oq

an]

eo

nA FPO KM EO OO DW IT AN HPO WB O

wo

Oo

&
OO O~? An B&W BHO VO COA

>
e)

FO DY

p

AO HAM CO OO COr Qn

a

be

81
HF.
4 PER-CENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.
z | Colog Nz A

Log M, A Colog M, A x

ec ES |S |, | Ce |

022 037 |g" 164 964 | 993 899 |5°835 036] 006 101

nd

as > BW Go r) rN) 7
OO OAM FO HO OO ON AN SPO KH BO WO GH AN HW HHO OO Or Qn © WH =

ir
Oo

pO bb =

949 732
"972 236

_"994 882
6°017 669

"040 602
"063 687
"086 940

"110 374
"133 999
157 814
"181 819

"206 OO!

"230 347
"254 844
"279 485
"304 269
"529 201

"354 292
“379 554
"405 O04
"430 658
"450 527
482 625
*508 968
"535 569
"502 443
"589 607

617 079
"644 876
"673 O19
"FOI 529
°730 431

"759750
"789 514
"B19 759
"850 523
‘881 850

"913 790

"946 391

"979 795

5°03 779

"048 669

022 203
022 357
022 504
022 646

022 787
022 933
023 085
023 253
023 434

023 625
023 815
024005
024 182
024 346

024 497
024 O41
024 784
024 932
025 091
025 262
025 450
025 O54
025 869
026 098

026 343
026 60!

026 874
027 164
027 472
027 797
028 143
028 510
028 go2
029 319
029 764
030 245
030 764
031 327
031 940
032 601
033 314
034 974
034 890
035757

"158 863
"150 888
"141 387
"130 735

"11Q 307
"107 394
'095 337
"083 471
"O72 I15
"O61 322
"050 464
"039 192
"027 177

"014 042

3°999 625
"984 015
‘967 808
"951 196
934 519
"918 146
"902 138
"886 555
"B71 393
"856 360

"841 404
"826 828
"812244
797 743
"783 328
"769 OO!
“754 793
"740451
"726 261
"712 120
698 o15
‘683 947
"670 021
656 268
"642 693
"629 270
615 875
"602 325
588 493
"574291

992 025
G90 499
989 348
988 572
988 087
987 943
988 134
988 644
989 207
989 142
988 728
987 985
986 865
985 583

"841 137
‘849 112
"858 613
"869 265
880 693
"892 606
"904 663
"916 529
"927 885
"938 678
"949 536
"960 808
"972 823
985 958

984 390 4°000 375

983 793
983 388
983 323
983 627

983 992
984 417
984 838
984 967
985 134

985 334
985 416
985 499
985 585
985 673
985 702
985 748
955 810
985 859
985 895
985 932
986 074
986 247
986 425
986 577
986 605
g86 450
986 168
985 798
g85 282

015 985
"032 192
7048 804
"065 481

"081 854
"097 862
"113 445
°128 607
"143 640
"158 506
"173172
"187 756
"202 257
"216672

"230999
"24.5 207
"2.59 549
"273 739
287 880

"301 985
"316.053
"329 979
"343 732
"357 397

"370 730
"384125
"397 975
"411 507
"425 709

007 975
009 501
010652
O11 428

O11 913
012057
o11 866
O11 356
010 793

010 858
O11 272
O12 015
013.135
014417
o15 610
016 207
016 612
016677
016 373

016008
O15 583
O15 162

015 033
o14 866

014 666
org 584
Ol4 50!
O14 415
O14 327
org 298
O14 252
O14 190
O14 14!
Old 105
014 068
013 926
013 753
013 575
O13 423

O13 395
O13 55°
013 832
Ol4 202
014718

we bt
Oo Sw AN F&O DO

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LSS)

So
Oo OX AN PO DH O HO ON AN O&O D&O

> +
°

OO O~3 Qh HOD we

tn
Oo

fF G DN Hw

55 3°849 151
"824 111
"798 713!
"772 950
-746 820

Lo, nN
WO Wr~ IAN PON HO OHO ONAN HPO HHO Oo cnr A

~

~~

foe)
1)

CO
WO con~ Qo fF © Db

i)

beng

Log D,

"720 214! °
692 773
"664 357
"634 785
603 gI0

°571 624
"538 047
"503 205
*466 QO!
"428 969
*389 206
"347 399
"302 504! °
"254 339) °
"202 470) °
"146 499) °
"086 184] °
"021 437] °
"953 908} -
883 792] °
"B11 567 °
"738 134) °
663 813} °
"585 459) °
*500 250
"406 O47
"299 832

179 249

"044 762
"898 003
"742.771
"586 052
“432 933
*301 261
"198 207

"O9T 95°

( Log. pr)

89

HF.

4 PER-CENT.

Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences—(continued).

Colog D

1 974 gOore 150 849] 0°025 040 4915 S74 963 311) 5
"025 398| °878

"025 757
"026 136] °
"026 606

"027 441
7028 416
"029 572
"030 875
"032 286) °

°033 577
"034 842
"036 304| °
"037 932
"039 763

"041 897
044 805
048 165
‘051 869
"055971
"060 315
"064 747
"007 529
“O70 116] *
"072 225

"073 433
"074 321
"078 363
‘085 200| °
"094 203
"106 215
"120 583
"134 487
"140 759] °
"155 232
"156719
"153 119
"131 672
"103 054] °

‘974 602

"974 243) °
"973 864) °
973 394| °

972 559| °279 786
"971 584] °307 227
"970 428) °335 643
"969 125} ° !
"967 714] °396 ogo
"966 423] °
"965 158] °
"963 696) -496 795
"962 068) °
‘960 237

"958 103} ‘610 794
"955 195} °

"995 797| °
893 785) °593 953
879.417) “7
"865 513] °
"853 241/ °95
"844 768)/2°101 997
843 281] °
846 881) -
"868 328) °567 067
896 946) °698 739
‘893 743| “80 793
"881 375) "908 o50
"854 448)1'026 675
"803 990] 172 227
681 936] *368 237

‘686 301

A
7 |(Colog vpz)

"118 625
"145 552

Log Nz

nen d © or oft | &

A

“I ~r
BB & db = OO O~r An BRO PY HO LW CO~r Qn

co
© Ost Aan

2) 2)
OO cor Air APO WO HH O

83

HF.
4 PERCENT.

arithms and Co-logarithms of D,, N,, and M,; with
md their Deferencos Teortinedy

a | Coleg] Tr

984 881 |4°440 427] 015 119

in
wn

BD © rr An

QZ pow ps

~
Oo CONTA PW HH CO OO O~N A

~I

© wmrak kh & DO

5'061 466

"137 923
"217 557
*300 250
"386 192
"475075
"569 205
"667 707
"772 586
_ “884 999
3°005 788

135 459
"273 598
"418 498
"567 593
"717 047
"865 583
2°O10 004
"151040
"295 725
"458 035
648 341
"883 771
¥°197 750
686 301

544454

"529 O17
"513 282

‘497 240

480 758
"403 246
"444 523
"424 346
"402 542

"379 003
"353 975
"327 554
"299 509
"269 661

"237 791
"203 547
"165 947
"124 468
"078 736

028 302

2°972 916
"912 507
"849 207
"783 307

"715 465
646 869
"578 013

"335 975
"233 740
"IIS 977
1'983 100
"836 696
“680 674
"522 589
"368 085
"237 902
"141 039
"042 LIQ
0°931 455
"794 943
"606 296
"296 666

984 563
984 265
983 958
983 518
982 488
981 277
979 823
978 196
976 461

974972
973 579
971955
970152
968 130

965 756
962 400
958 521
954 268
949 506

944 O14
939 591
936 700

934 100
932 158

937 494
931 144
927 227
g2z0 118
glo 623

897 765
882 237
867 123
853 590
843 978
841 915
84.5 496
869 817
903 137
gor o80
889 336
862 588
812 253
690 370

"455 546
"470 983
"486 718
"502 760

"519 242
"536 754
"555477
"575 954
"597 458
"620 997
*646 025
*672 446
°700 401
°73° 339
"762 209
796 453
"834 053
"875 532
"921 264

"971 698
3°027 084
"087 493
"150 793
"216 693

"284 535
"353 131
"421 987
"494 766
"574.048
"664 025
"766 260
"884 023
2°016.900
"163 304

"319 326
"477 411
‘631915
‘762 098
"858961

"957 881
1068 545
°205 957
"393 794
°793 334

015.437

015 735
016 042
016 482

O17 512
018 723
020177
021 804

023 539

025 028
026 421
028 045
029 848
031 870

034 244
037 600
O41 479
045 732
05° 434
055 386
060 409
063 300
065 goo
067 842

068 596
068 856
072779
079 882
089 377
102 235
117 763
132 877
146 404
156022
158 085
154 504
130 183
096 863
098 920
110 664
137 412
107 747
309 630

CO Cc ~ ~T
Spwvond © onan

© On An #£$O Bm CO OO CHr~3 Qn ABW WB | CO W/O COT A

)

OO cm? Qin PG DN om

lo) bana

&

+b as bo © w
BoP HMO LO CONTAN P&WH HO LO CONTA FH HNO LO ONO £WO HHO OO ON AN $W Db HO

oy

az

19°373 1
IQ’2l 5|°
"228 290
"233 484
"238 342

"243 003
"247 577
"2.52 209
"257 O45
"2062 236

19°064 5
18°929 4
18°803 1
18°681 9
18°563 0
18°442 6
18'3168
18°181 9

18°0366
17°887 9
17°740 0
17°597 9
17°463 8

17°3429
17°233 7
17°129 4
17°027 6
16°9240

16°813 9
16°696 1
16°569 6
16°434 1
16°293 !
161458
15991 4
15°831 4
15°065 3
15°493 0
15°3140
151290
14°937 3
14°738 4
14°532 2
14°318 5
14°096 8
13°865 3
13°6229
13°369 0
13°103 5
12°827 5
12°543 I

11°954 5

A;

"216 420

222 636

"267 824
"273 S43
"279 229
"284 731
"289 855
"294 506
"298 702
"302 716
"306 632
"310615

"314 852
°319 380
"324 244
"329 456
"334 881

"340 547
"340 484
“352 O40
*350 026
"305 054

"372 537
"379 955
387 027
"394 676
"402 608
"410 829
"419 354
*428 260
"437 582
"447 345
"457 558
"468 174

"479 110
12°2519]°

490° 313

‘501 752

84

HF.
4 PER-CENT.
‘alues of Annuities, and Single and Annual Premiums for
Assurance of a Unit.

Ws

‘010 623
“OLI OLS
“ori 378
"O11 716
"O12 036

"O12 346
"012 655
"012 972
"O13 397
"013 671
"OI4 069
"O14 482
"O14. 900
"Ol5 311
"015 O99
016.056
"016 382
"016 698
"O17 009
"O17 330
‘017 675
"018 048
"018 455
"018 897
"019 365
"o1g 862
"020 392
"020 951
"O27 543
"022 170
"022 835
"023 539
"024 284
"025 077
025 O21
026 819
027 778
"028 809
"029 925
031 133
"032 443
033 858
035 377
‘037 000
038 732

z

*
Oo cn~? Qn

bond

“TI ~r NAN
OO wom AQ #+$&H SB CO YO COr~? Qn

oc
Oo

oe)
Oo cnr Atnr & BW =

‘©
oO

Oo
© OI An PW DH &

az

11°652 6

11°344 2

As

"513 361
"§25 222

11027 4| "537 408

10°'701 2
10°3650
1o°019 8
9°673 4
9°327 5
8°984 7
8°646 7
8°3140
7°982 3
7649 1
7°3160
6°983 7
6653 3
6°3271
6°0147
5°720 2
5°445 8
5°194 9
4°968 9
4°7678
4°569 8
4°579 5
4°161 3
3°9279
3°6610
3°3849
3°118 6

2°874.0
2°670 4
2°5249
2°441 4
2°4230
2°4640
2°5348
2°606 3
2°520 4
2°206 8
1818 5
1°389 6

"942 9

"480 8

‘000 0

"549 9.54
"562 885
"576 161
"589 486
‘602 790
015 973
"628 973
"641.770
054 517
607 343
680 155
"692 935
"705 643
-718 188
"730 202
741 531
752 083

"761 734
"779 427
"778 163
‘785 776
"793 442
"801 489
"810 466
"820 730
831 348
"B41 592
"850998
858 832
"864 425
"867 637
"868 347

"866 768
"864 046
"861 296
864255
876 662

‘891 597
"908 og1
"925272
"943 048
"GOI 539

Ws

"049 573
"042 548
"O44 682
"047 000
"049 528
052 284
05,5 230
058 368
"061 692

"065 201

"068 904
072 869
"077 158
“081 789
086 704

"092 201
098 018
"104 095
"IIO 344
°116677

"122 961
"129 074
"134.916
"141077
"147 740

"155 289
"164 465
"176 084
"189 592
"204 339
"219 666
"233 991
"245 231
"252115
"253 681

"250 220
°2.44 440
"238 830
"244 874
‘273 378
"316 340
°380 O11
"476 226
‘636 863
"961 539

HF.

FOUR AND A ILALF PER-CENT.

rs) fund fund

wm hObD & O WO OD Qn APO YW HO

tS]

G2

&o
© os? Qn Pwo BHO WO CON A

in poe nD

>

—n
hObHeE O OO ON A

86

HF,
4} PERCENT.
Commutation Table.
D,; Ns S, M, R,

64 392°8 [1148675 [19 196 516 [12.155°4 1334 185°9
614264 |1087248 [18047842 |r11 Q61'9 | 322 030°4
58 533°6 |1028715 {16960594 |11 714°3 |310068'5
55 725°2 | 972989°5 |15931879 |11 426'5 | 298 354°2
530118 | 9199777 114958889 [11 112°8 |286927°8
50 402°5 | 869575°2 |14.038912 |10 786°2 |275815°0
47 :902°2 | 821673°0 |13 169 336 |10456°4 |265 028°8
45 516°3 | 7761567 |12 347663 |10133°2 |254 572°4
43 248°3 | 732908'4 [11571 507 | 9 825°30) 244 439°2
411004 | 691 808'0 |10 838 5908 | 9 539°761234 613'9
39 067°2 | 652740°8 110146790 | 9 276°46)225 O74°1
37127°8 | 615613°0 | 9494050 | 9 019°34/215 797°7
35270°0 | 580 342°9 | 8878437 | 8 760°39) 206 778'3
33 483°8 | 5468591 | 8298094 | 8 492°96) 198 017°9
31759°2 | 515099°9 | 7751234 | 8210°28) 189 525°0
30092°5 | 4850074 | 7236135 | 7911°15/ 181 314°7
28 485°2 | 456522°2 | 6751127 | 7 599°76|173 403°5
269484 | 429573°8 | 6294605 | 7 289°58! 165 803°8
25 483°3 | 404090°5 | 5 865031 | 6984'89/158 514°2
24092'°9 | 379997°5 | 5460941 | 6 691°92) 151 529°3
22779°9 | 357217°6 | 5080943 | 6416°38) 144 837°4
21 540°6 | 335677°0 | 4723726 | 6158'07| 138 421°0
20371°7 | 315 305°2 | 4388049 | §916°74) 132 263°0
19 268'9 | 296036°3 | 4072743 | 5 691°20/126 346°2
18 224°2 | 277812°1 | 3776707 | 5476°26/120655'0
17 235° | 260577°0 | 3498895 | 5271°86) 115 178°8
16298'9 | 2442781 | 3238318 | 5 077°90) 109 g06'9
I5411°5 | 2288667 | 2994040 | 4 892°30/ 104 829°0
14.5702 | 214296°5 | 2765173 | 4714°68] 99936°71
13772°8 | 200523°7 | 2550877 | 4544°721 95 222°03
30171 | 187 5066 | 2350353 | 4382°08) 90677°31
12 300°2 | 1752064 | 2162846 | 4225°78] 86295°24
11620°5 | 163 585°9 | 1987640 | 4075°74| 82 069°46
109762 | 152609°7 | 1824054 | 3931°86] 77 993°72
10 365°4 | 1422442 | 1671445 | 3 793°74| 74.061°87

9 786°37| 132.457°8 | 1529200 | 3 661°02! 70 268°13

9 237°41| 123 220°4 | 1396743 | 3 533°48) 66607"12

8717°96| 114.502°5 | 1273522 | 34t1°82) 63 073°63

8 226°73| 1062758 | 1159020 | 3296°00|) 59 661°82

7762°21| 98 513°54| 1052744 | 318574) 56 365°82

7 322°78| 91 190°76] 954230°3} 3 080°57} 53 180°07

690616] 84284°60]} 863 039°5] 2979°29] 50099°51

6 509°92/} 7777467) 778754°9| 2880°44) 47 120°22

613229} 71642°39| 700980°3| 2 783°14] 44 239°77

5§771'94| 65870°44| 629 337°9| 2 686°87| 41 456°6.

wo ~ b=
0 © G1 An APO WP = OC

we

> > &o Oo
ho bD eM O OO Or Qn APO Be CO OO Cr? An BO WH mH

ony
ko bwm CO O OXI’ AH

87
HP.

4} PER-CENT.

Commutation Table—(continued).

rn Lr

OQ pw pn

~
ROW = CO OO Cr A

CO ~
YO Cn~7 AG P&H Dm O W/O CON AN

co

oO ow ak pow dS

3 937°65
3 678°85
3 429°37
3 188-30
2 955°29
2 730°44
2 515°20
2 310°I9
2 114°76
1928°61

1751°46
1 582°78
I 420°79
1 265°56
1 117°72

977°861
846°992
726°188
618°640

37 925°03
34 246°18
30 816°81
27 628'52

397 121°9
351 052°6

309 189'9
271 264°9
237 018°7
206 201'9

24 673°22
21 942°78
19 427°58
17 117°40
15 002°64
13 074'02
LI 322°57

178 573°4

153 900°2

131 9574
T12 529°8

523°88y

441°500
371°035
311180
258°563
211°486

169°432

132°038
99°548 0
72°687 9
51°596 5

35°917 5
24°917 5
17°4302
12°809 9
10°05 5 6

7°835 6
5°934 2
4°224 0
2°6769
1°280 8

9 739°789
8 318°996
7 953°434
5935°719
4.957858
4 110°866
3 384°677
2 766°037
2 242°148
1 800°648
I 429°613
1 118°433
859°871
648°384
478°952
340°915
247°367
174°679
123°082
87°165
62°247
44°817
32°007
21°952
I4'116
8°182
3°958
1°281
"000

95 412°42
80 409'78

67 335°76
56 o13°19
46 273°40
37 954°41
30 900°97
24 905°25
20 007°39
15 896°53
12 511°85

2 134°95
2 045°72
19 54°65
1 861°26
1765°55
I 667°96
T 570°30
1473°59
1 377°64
1 282°57
I 188°46
1095°20
1 001°38
907°327
813°979
722°256
633°496
549°166
472°889

9745815

7 503°667
5 793°O19
4.27 3°406
3 154973
2 295°102

1646°718
1 167°766
820°853
573°485
398°806

275°] 24
188°559
126°312
81°495
49°488

27°530
13°420
5°239
1°281
"000

494°777

344°949
293°495
249°617
210°400
174°458

I4I°511

III413
84°609 I
62°035 8
44°0744
30°617 3
21°1640
14°749 7
10°8800
8°677 3
6°8g0 3
5°326 3
3°8717
2°500 5
1°225 7

38 769°77
36 178°94
33 682°98
31 280°10
28 968-29
26 745°59
24 610°64

22 564'92
20 610'27

18 749°01
16 983°46
15 315°50
13 745°21
12 271°62
10 893'97

9 611°403
8 422°941
7 327°739
6 326°362
5 419°035
4 605°056
3 882°800
3 249°303
2 700°138
2 227°249
1 822°472
1477°524
1 184°028

934411

724°OUI

549°552
408'041
296°628
212°019
149°983

105°909
75°291
54°127
39°378
28°498
19°820
12°930
7°60.
3°732
1'226

oN
O cnr Qn

5

co ro) ~ ~ ro
ww ~O bY

OO Or Qn BRO HY wt O WO W~7 Qin POP FO WO CONT A

© rr ak pow Sd

88

HF.
4} PERCENT.
Logarithms and Co-logarithms of D,, N,, and M,; with

their Differences.

A

A A
Log De | (Log epz) Colog Ds |(Colog ep;)| “8 Ne
4" Bon 837/1°979 518)5°191 163] 0°020 482 |6°060 1971976 132
"788 355] °9790950| ‘211 645) "020950} °036 329975 966
"767 405] 978.647) °232 595] 021 353] ‘012 2951975 813
"746 052} °978 321) °253 948] '021 679 /5°988 108/97 5 669
"724 373| °978.079| ‘275 627| °021 921} 963 7771975 53°

"702 452] ‘977 904) ‘297 548) °022 006| "939 307/975 392
"680 356) °977 811] “319 644] °022 189] “914 6991975 253
658 167| 977 802] °341 833} 022 198] *889 9501975 100
"635 969] °977 877) 364.031] °022 123] °865 0501974 936
"613 846) 977 967| *386 154] "022 033} 839 9861974 755

591 813] °977 886] -408 187) *022 114] °814 7411974 567
"569 699] °977 707| *430 301) °022 293] *789 308.974 377
547 406| 977 420] “452 594| 022 571| °763 6851974 190
"524 835] °977 935) 475 165) "022.965| °737 8751974 O17
"501 870] °976 588) *498 130] ‘023 412] °711 892/973 856

"478 458] 976 162) -521 542) °023 838) 685 748.973 71

454.620) "975 914 °545 380] °024 086] °659 4621973 576
"430 5341 °975 721) °569 406) °024.279) °033 0381973 441
"406 255) 975 635} °593 745| °024 365] *606 479973 302
381 B90] °975 662) 618 110] *024 338] °579 7811973 152

"357 5521 975 707| 642 448) °024 293] "5529331972 989
"333 259) °975 799| ‘666 741] ‘024 231] “525 9221972 809
"309 028] °975 830] *690972| 024170) "498 7311972 O14
"284 858] °975 791) “715 142) 024200] °471 3451972 406
"260 649] °975 754) °739 351] 024236) “443 7511972 185

"236 413) "975 746] °763 587) °024254| “4159361971 949
"212 1599) 975 685) °787 841) ‘024 315] °387 8851971 698
"187 844) °975 621) “812 156] 024 379} °359 583/971 432
"163 465) °975 557) °836 535] °024 443] “331 015/971 151
"139 022] °975 491! *860978] 024 509] “302 166/970 851

"114 513] °975 400] °885 487] *024 600] °273 017/970 533
"089 913] 975 312) °g10 087] *024 688) 243 5501970 196
"065 225] 975 228) -934775| 024772] 213 746\969 836
"040 453] °975 135] "959 547| °024865| *183 582/969 453
"O15 588) 975 033] y84412| 024 967| 153 035/969 043

3°990 621] "974 orale 009 379| °025071| °122 078/968 605
"965 559] °974 865] °034.450| 025 135] ‘090 683/968 132
"940 415] "974 812] ‘059 585] *025 188] 058 8151967 O19
"915 227] °974759| ‘084 773} '025 241] °026 434/967 062
"B89 986] °974 690) ‘110 O14] *025 310 14°993 4961966 455

"B64 676] °974 561) “135 324] "025 439) 9599511905 797

"839 237] °974 339| “160 763] *025 661) 925 7481965 ogo

"813 576| °974.046] *186 424] °025 954} "890 8381964 332

787 622! *973 700] ‘212 378] °026300| °855 1701963 521

"761 322] ‘973 266| °238 678) 026734! °818 6911962 656
i

be we = =
© CO~31 Aw PO HW | O

&o
© OX An AHH PHO © CO™ Aw POH HO

9

~ pow nd

Ca
hwo N HO W/O CONF AN

Len a!

S)
nN FOoOKK CO DO GAN $Y O

ws

&9
© ONT An POD CO HO Or~ a

&o

pp
ro)

= PO O

tr
khROo Dw O OO COrT QMX

89

HF.
4) PERCENT.

Logarithms and Co-logarithms of D,, N,,. and M.; with

their Differences.
2 |CologN,| A

Log M,; A Colog M,

es | ee | ee | eens | EE | een

023 868 4°084 77° | 993 931 |5°915 230] 006 969

A @

=

OS cox~?r Qtr © Wb mw

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0 OONT AN AW HHO

&o

I
2
3
4

35
6
|
8
9

3

© OTA BOB

"903 671
"987 705
"o11 892
036 223

"060 693
"085 301
"110050
°134.950
"160 O14

*185 259
"210 692
"236 315
°262 125
288 108

"314252
"340 538
"366 962
"393 521
"420 219

"447 067
"474,078

024 034
024 187
024 331
024 470
024 608

024 749
024 900

"077 801
"068 715

"057 913

045 822

"032 868
"org 382
"005 74/7

025 064|3°992 346

025 245
025 433
025 623
025 810
025 983
026 144
026 286
026 424
026 559
026 698
026 848

027 OI!
027 191

"S01 269/027 386

"528 655
"556 249

027 504
027 815

584 064/028 o51

"612115
"640417
*668 985
"697 834

"726.983
°750.450
"786 254
816418

"040049
"074.252
"109 162
°144830
"181 309

028 302
028 568
028 849
029 149
029 467
029 804
030 164
030 547
039957
031 395
031 868
032 381
032 938
©33 545
034 203
034910
035 668
036 479
037 344

"979 537

"967 382

"955 175

"942 523

"929 059

"914 358

"898 240
"880 800
"862 702
"844 160
"825 551
"B07 290
°789 444
"772 083
°755 203
"738 484
"721 964
"705 684
"689 513
"673 453
"657 5°7
*641 680
"625 907
"610 206
"594 598
"579 007
*563 602
548 203
"532 986
"517 987
503 211
488 631
"474113
"459 459
"444 535
"429 240

990 914
989 198
987 909
987 046

986 514

986 365
986 599
987 191
987 845

987 793
987 348
986 536
985 299
983 882
982 560
981 go2
981 458
981 391
981 739

982 154
982 639
983 120
983 281
983 480

983 720
983 829
983 940
984 054
984 173

984 227
984 299
984 392
984 469
984 535
984 601
984 783
985 001
985 224
985 420

985 482
985 346
985 076
984711
984 193

"922 199
"931 285
"942 087
954 178
"967 132
980 618

_ 994 253

4°007 654

"020 463
"032618
"044 825
"057 477
"070 941
"085 642
"101 760
"119 200
"137 2098
"155 840
"174.449
"192 710
"210 556
°227 917
"244 797
261 516
278 036
"294 316
"310 487
"326 547
"342 493

"358 320
°374 293
"389 794
"405 402
°420 933

"436 398
"451 797
"467 O14
"482 013
"496 789

"511 369
"525 887
"540 541
"555 465
"579 754

009 086
010 802
O12 OO!
012 O54
013 486
013 635
O13 401
o12 809
O12 155

O12 207
012 652
013 464
014701
016 118

O17 440
018 098
018 542
018 609
018 261

017 846
O17 361
016 880
016 719
016 520

016 280
O16 171
016 060
015.946
O15 827

015773

O15 701 |-

o15 608

O15 531
O15 465

O15 399
O15 217
O14 999
org 776
o14 580

org 518
O1g 654
O14. 924
O15 289
015 807

» bet
© COrF Qt > © WH HO

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© Go
Oo C3 An P&H WHO OO CN A

aN

a fF G YW mH O

-&O We oO OW CO~ A

- 90

HF,
4} PERCENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

~
tn

sy

~
Oo wom? Ot PHONO OO CO™ A

c

CO
© cn~7 Own P&H BO & O

pep nd 0 Ord

"707 405
"679 985
"052 145
"623 926

"595 237
"505713
"535 214
"593 559
"470 601

"436 233)"

"400 573
"303 O47
325 261

"285 245
‘243 400

"120 698

1°998 033

"861 462
"712 620

"5.55 306
"396 504
"241 301
"107 546
"002 409;

0804 070

773 362
"625 727
"427 634
"107 487

"972 520
"972 160
971 781
"O71 311

"970476
"969 501
"968 345
967 O42
"965 632

"964 340
"963074
"961 614
"959984
"958155
"956.020

"953 11
"949 752
"946 048
"941 946
"937 603 |3°009 723
"933 169
"930 389| "138951
"927 801
"925 692] 280 761

"852 365

°292 535:
"320 015)
"347 855)
"376 074

"404 763
"434 287
464, 786
"496 441
"529 399
"563 767
"599 427 |
"636 353
"674 739:
"714755
"756 600
"800 580
"847 469
"897 717
"951 669

"072 120

‘208 562

"924 484] °355 059
"923 596] °430 585
"919 555| “506 989
"942 716| °587 434
"903 714] °674718
"891 702
"877 335 |_°879 302
"863 429 |2°001 967
"851 158] 138 538)
"842 686

"841 198) -444 694:
"844 797 | *603 496
"866 245] °758 699
"894 863 | *892 454
"891 661

"879 292 |1°105 930

"775 004

"287 380;

"997 591

"226 638

"801 907) "374273
"679 853| °572 366
"892 513

"027 480
"627 840
7028 219
028 689

"029 524
"030 499
"031 655
"032 958
"034 368
"03 5 660
"036 926
038 386
"040 016
"O41 B45

"043 980
"0.46 889 |3°988 550 | 931 521
"050 248
053.952
058 054
"062 397
"066 831
"069 O11
"072 199
"074 308

"075 516
7076 404
080 445
"087 284
"096 286

"108 298
"122 665
"136571
"148 842
"157 314
"158 802
"155 203
"133.755
"105 137
"108 339
"120 708
"147 635 |0°912 846 | 684 600
"198 093| °597 446 | 510 041
"320 147

743 073 | 960 719
"793 792 959 619
"663 4111958 416
621 827 957 099

"578 926 | 955 686
*534 612/954 176
488 788 952 370
"441 358.950 868
°392 226 ' 949 066

"341 202 (947 127
288 419 945 019
"233 438 | 942 730
°176 168 | 940 241
"116 499 | 937 536

"053 945 | 934 605

"920 071 , 928 330
"848 401 |925 072
"773 473 |921 821

"695 294/918 639
"613 933 (915 584
"529 5U7 | O12 341
"44.1 858 | 908 806
"350 664 | 904 765

"255 429 | 899 789
"155 218 | 893 392
048 610 | 885 823
2°934 433 | 877 400
811 833 | 868 459

"680 292 | 859 930
"540 222 | 853 119
"393 341 | 848 899
"2.42 240

"090 195 | 850 146

1°940 341 | 853 779
"794 120 (857 323
"651 443 '853 804
"505 247 836 217
"341 464: 808 246

"149 710, 763 136

847955

"107 487

l
A A
Log Ds | (Log up,) | Col Da (Colog rp)| WBNe | A | 2

3°734 5§88]1°972 877 |4°265 4120°027 123 |4°781 347°961 726) 5
|
|
{
|
j

OO DY OT AHHH ODVIAR AOBHS

~r

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CO

co
Oo OX AN POH |= O

1 cwx~r Ah we

Logarithms and Co-logarithms of D,, N,, and M,; with

z
53 5°218 653
*256 927
’ "296 208
8 | °336 589
9 | 378173
6o | *421074
I 1.465 388
2 | 511212
3 | 558642
4 | °607 774
65 | °658 708
61-711 581
7 | -766 562
8 | 823 832
9 | °883 591
70 | "946055
I Iq°oIr 450
2 | 7079929
3 | “151599
4 | °226 527
75 | °394 706
6 | °386 067
7 | 479 483
8 | °558 142
9 | °649 336
80 | °744 57!
1 | 844 782
2 | “951 399
3 |3°065 567
4 | °188 167
85 | 319 708
6 | 459778
7 | °606 659
8 | °757 760
9 | “909 Bo5
90 |2°059 059
I | ‘205 880
2 | °348 557
3 | “494753
4 | 658 536
95 | 850290
6 | 087 154
7 | °402 554
8 | °892 513

9

038 274 |3°413 439

039 281
040 381
O41 584

042 gor |

044 314
045 824
047 430
049 132
050 934
052 873
054981
057270

059 759
062 464

065 395
068 479

131 541
140070
146 881
151 101
152 045
149 854
146 221
142677
146 196
163 783
191 754
236 864

91

HF,
41 PER-CENT.

their Differences.

Log M,

"397 237
"380 734

"363951

346 882

"329 388
"310 846

"291 069

269 807
246 880

"222 185
"195 981
"168 377
"139 137
"108 080

"074.985
"039 494
"O00 597

2°957 704

=

315 400].

489 959

"910613

858 691
"B01 744
"739 793
‘074 759
"607 216

"537 754
"4607 O01
"397 275
"323 046
"241 692

°150 791
"046 936
"927 417
"792 642
644 187

"485 967
"325 597
*168 782
"036 629

0'938 386

"838 237
"726 429
587 go2
"399 064
088 371

A

983 798 |4°586 561

983 497
983 217
982 931
982 506

981 458
980 223
978 738
977 973
975 395

973 796
972 396
970 760
968 943
866 905

964 509
961 103
957 167
952 849
948 078

943 953
937 959
935056
932457
930 538

929 847
929 674
925771
918 646
909 099
896 145
880 481
865 225
851545
841 780
839 630
843 185
867 847
901757
899 851
888 192
861 473
8r1 162
689 307

Colog M,

"602 763
619 266

636 049
653 118

"670 612
689 154
"708 931
°739 193
"753 120
"777 815
"804 019
831 623
*860 863
"891 920

"925 O15
"960 506
_ 999 493
3°042 236
"089 387

"T4I 309
"198 256
"260 297
"325 241
"392 784
"462 246
"532 399
"602 725
676 954
°758 308

"849 209
953 064
2°072 533

"207 358

"355 813

"514 033
‘674 403

831218
"903 371

T'061 614

"161 763

"273 571
"412 098
‘600 936
"91x 629

A

016 202
016 503
016 783
017 069
017 404
o18 542
01977

O21 262
022 927
024 695
026 204
027 6o4
029 240
031057
933 995
03549!
038 897
042 833
047 151
051922

056 947
062 04)
064 944

007 543
069 462

070153
070 326
074 229
O81 354
090 go!

103 855
119 519
134775
148 455
158 220
160 370
156815
132153
098 243
100 149
rrr 808
138 527
188 838
310693

i
© Orr Qu

GO ~“ a I “A an
OO ONAM POY HO OO ON AN $& Bm OG

Cc
OO CONIA PW WHO

© wr ak pwwond

92

HF

41 PERCENT.

Values of Annuities, and Single and Annual Premiums for

Assurance of a Unit.

8

> &o wo wo wo b= bs
0 OO ONAN SPO YH MO OO GO Ann POH & CEC HO GO~ Qn HW WH HO

aN

ROH MO DO CONAN PWD &

17°838 6
L7°700 0
17°574 8
17°460 5
17°3542

17'2526
17°353 1
17°052 3
16°946 5
16°832 1
16°708 I
16°5809
16°454 3
16°332 0
16'2189

16117 2
16°026 6
15°9.40 6
15°857 1
15°7721
15°681 3
15°583 4
15°477 6
15°363 4
15°244 1

I5°11g0
14°987 4
14°850 4
14°707 9
14°559 4
14°404 7
14°244 2
14°077 3
13°903 7
13°7229

13°5349
13°339 3
13°5341
12°918 3
12°691 4

12°453 0
12°204 3
119471
11°682 8
V1'g12 2

"188 770
"194 736
"200 129
"205 O51
209 628

"214 002
-218 286
"222 628
227 184
232 109

"237 449
"2.42 927
"248 380
‘253 643
2.58 517
"262 895
"266 796
"270 501
"274097
"277 755
"281 668
285 881
"290 439
"295 356
"300 493
°305 880
"311 548
"317 446
"323 585
"329 978

"336 641
"343 553
"35° 730
"358 216
‘365 999
"374 004
"382 519
"391 355
"4.00 645
“410 417
"420 683
"431 396
"442 469

453 850

"$05 504

"OLI 724
"012 025
"O12 332
"012 659
"013 016

"O13 409
013 818
"O14 230
"014 634
“OIS O14

"o15 358
"O15 669
"015 968
"016 260
"016 561

"016 885
"017 239
"017 626
"018 o50
"018 499
018 976
019 487
"020 028
"020 600
"021 208

"021 853
"022 537
"023 263
"024 035
024 859

"025 738
"026 676
"027 689
028 785
"029 976
"031271
"032671
"034.175
035 785

037 504

in
Oo COnr Qu

puwnAs

oo oo) ° ~“ ~ A
OO O~N AN POLY = GQ DO Cn Qn BPO HMO WO Or an

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oO

©
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11°1367
10°854 5
10°563 5
10'262 8

9°9518

9°631 4
9°308 9
8°986 2
8-665 6
8-348 8
8°036 4
77241
7°499 5
7094 3
6°779 0
6°464 7
6°153 6
5°855 2
5°5734
5°3106

S070 1
4°853 5
4°660 9
4°4712
4279 8
4°078 5
3°853 0
3°594 2
3°325 6
3°065 8
2'8268
2°627 4
2°4849
2°403 I
2°385 5
2°4268
2°498 I
2°5712
2°498 6
2°183 0
1801 5
1°3787

"9370

"478 5
*9000 O

"477 395
"489 522
"502 052
"514.999
"528 392

"542 189
"556 075
"569 974
"583 778
"597 419
610 875
"624 323
637 867
"651 444
*665 020

678 555
"691 950
"704 801
"716 937
"728 252
*738 608
747937
"756 230
‘764 400
"772 639
‘781 310
"791 O17
"802 165
"813 731
"824.916

"835 209
"843 797
"849 933
853 454
"854 214
"852 434

"849 363
846 215

849 342

"862 933

879 360
897 567

"916 591

936 334

"956 938

"010 020
"O10 414
"010774
‘011 108
‘OII A21

"039 332
"O41 294
"043 417
"045 726
048 247

"05° 999
"053 941
"057 076
“060 398
"063 903
"067 602
"071 563
075 851
“080 482
"085 489

"090 903
"096 727
"102 813
"109 067
"115 402
"121 679
"127776
"133 589
"139715
"146 339

"153 847
"162 904
"174 605
"188 121
"202 889

"218 252
"232618
"243 891
"250 785
"252 317
248 755
"242 804
"236.953
"242 705
"271 106
"313 886
"377 328
"473 213
"633 314
956.938

HF.
FIVE PER-CENT.

= bt
OW o~ Qn #$O® WB m O

SS

Go

°
I
2
3
4
5
6
7
8
9
fo)
I
2
3
4

>
WO GON? QKi

%

pops

jw
b&Owbm OO OO CON A

58 284°3
55 275°0
52 372°4
49 58570

46919°9
44 380'0
41 968°7
39 687°6
37 536°9
35 510°2
33 586°6
317541
30002°4
28 321°6

26 707° 5
25 160°6
23 689°8
22295°2
20 978'4

19 740°7
18 577°8
17 486°1
16 460°7
15 494°1

14 583°4
13 7256
12 916°4
12 153°2
Il 433°4
10 754°6
10 113°9
9 509°52
8 939°51
8 401°86
7 894°71
7 416°38
6 966°00
6 542°18
6 143°39
5 768°00
5 413°94
5 979°01
4 761°60
4 460°4

OI 391°3 [1013 415°

955 t30°8
899 8558
847 483°4
797 898°3
75°.978'5
706 598° 5
664 629'8
624 942°!
587 405°2

551 895°
518 308'5

486 554°4

450 552°0
428 230°4

401 522°9
376 362°3
352 672°4
3393772
309 398'8
289 658°2
271 080°3
253 594°3
237 133°6
221 039°5
207 056°!
193 330°5
180 414°!
168 260°9
156 827°5

146 073°0
135 959°!

126 449°5

117 510°0
10g 108°2

IOI 213°5

93 797°08
86 831°08
80 288-90
14.145°5!
68 377°50
62 963°56
57 884°55
53 122°95
48 662°49

16 006 094°
14.992 679°
14 037 549°
13 137 693°
12 290 209°

II 492 311°
TO 741 333°
10 034 734°
9 370 104°
8745 62°

8157757"
7 605 862°
7087 554°
6 600 999"
6 144 447°

5716217
5 314 694°
4 938 332°
4 585 659"
4255 282°
3 945 883°
3656 225°
3 385 145°
3 131 55°
2 894 417°

2672 777°
2465 721°
2272 391°
2 091 977"
I 923 716°
1 766 888°
1 620 815°
1 484 856°
I 358 407°
1 240 897°
I 131 788°
1030 575°

936 778°0
849 946'9
759 658°0
695 512°5
627 135°0
504 171"4
506 286'9
453 163°9

IO 210°C

10 026°5
9 792°60
9 522°14
9 228°69

8 94°69
8 619°13
8 32114
8 038°50
777780

7 538°47
7 305°88
7 072°74
6 833°12
6 581°03

6 315°55
6 040°50
5 797°83
5 501°26
524617

5 007°38
4 784°60
4577°46
4 384°78
4 202°04

4 029'09
3 865°76
3710°20
3 562°05
3 420°96

3 286°59
3 158°07
3 035°28
2918°10
2 80615

2 699°08
2 596°69
2 499°47
2 407°37
2320°11
2237°27
2157°87
2 080'75
2.005°19
1 930°80

261 430°3

251 220°2

241 193°7
231 401°)

221 879°0

212650°3
203 725'6

195 106°5
186 785°3

178 746°7

170 968'9
£63 430°
156 124°6
149 051°8
142 218°7
135 637°7
129 322°
123 281°6
117 533°8
112 O12°5
106 766°4
IOI 759°0

96 974°39
92 396°93
88 o12°15
83 810°10
79 781'Or
7591525
72 205°05
68 642°99
65 222°03
61 935°45
58 777°38
55 742°10
52 824'00

50017°85
47 318°77
44.722°08
42 222°61
39 81 5°24
37 495°13
35 257°86
33 100°00
31 019°25
29 O14°05

> Go Go ee 1S) i . =
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0 ON A pw KH do © OYIAN AWK HO OHO WIA BPW BH DO © ON An

‘6

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© cert Aw ff & DH & O

HF,
5 PERCENT. |
Commutation Table—(continued).

Dz B, R;
4.174'20 404 5OI'4 27 083'26
3.902°80 360 013°1 25 226°32
3 646°06 319 427°6 23 442°0I
3 403°38 282 488-2 21 728°59
3174°08 248 952°! 20 084'23
2 957°04 218 590°2 18 507°10
2 749°53 191 185"2 16.995°88
2 550°87 166 529°8 15 55134
2 360°26 144 425°3 14 174°54
2177°35 124 681°0 12 866°88
2 002°II 107 L141 LI 629°73
1 835°50 91 549°29 10 464°15
1 677°86 77 819°96 9 369°831
I 528°61 65 76850 8 345°749
1 387°42 55 245°65 7 391020
1253°98 | 7 881°459) 46 Ir0°21 6 504°689
1127°81 | 6753°648| 38 228°75 5 685°734
1 007°57 5 746°080] 31-475°10 4 933°231

893°210 | 4852°870] 25 729°02 4247°266
785°107 20 876°15 3 627°678
683°600 16 808°39 3 073°660
589°293 13 424°23 2 583°764,
502°838 10 629°36 2155°622
426°328 8 337°322 1 785°874
359°312 6 471616 1 468°691
301°363 4.965°222 I 198'222
2.52°059 3 760°191 968-592
2.10°390 2 807°218 773°916
173°983 2 064635 608°906
141°628 I 496°035 470°284
112°925 1 069°063 355°732
87°582 9 755036 263°139
65°717 5 528°552 190°511
47°7570 367°806 135°57
33°738 2 254816 95°475
23°374 1 175°565 67°17
16°138 4 119°688 47°517
11°235 3 79°949 34°039

8.2178 51445 2.4°696

6°420 2 31°160 17°836

4978 9 17°204 12°382

3°7528 8-408 8°063

2°658 6 3°274 4°733

1'6768 798 2°319

798 5 "760

~r wT] A (oy Car
Po WMO © WAI ON howe oO OO ON Qn Pwo HHO LW O™ AN

Le)
Oo cosr Qu

\O
Oo

0 mr an hb & O me

96

HF.
65 PER-CENT.
Logarithms and Co-logarithms of D,, Nz, and M,; with
their Differences.

A A
a | Log D, (Logeps) Colog D; (Colog vp) LogNz| A

T'977 445|5°211 893|0°022 555 |6°005 7871074 27

to 14°788 107

Lm

fp. &o & i) 4 =
0 WO Gr Amn S2OKH™ CO |W Cr Qn HBO Pe GC LO COr~? QM ADO YD

Ww WY

p.

rr
how m O OO Or Qn

"795 552

‘976977
"979 574
976 248
976 006

"975 830
975 738
975 13°
"975 804
"975 894

"975 813
975 934
"975 356
974 962
974 515

"974 089
‘973 840
"973 O49
"973 562
"973 589

"973 633
973 697
"973 756
973719
"973 690

973 674
973 611
"973 549
"973 484
‘973 418

°973 327
973 239
973 15514
"973 062
‘972 960
"972 856
"972 792
"972 739
"972 685
972 618
"972 487
972 266
"971974
"971 627
"971 193

‘234 448
257 471
"280 897] °
"304 649) °
"O24 170
"024 262
"024 270
"024 196
"024 106

"328 643

"352 813

"377975
"4Ol 345
425 541
"449 647

473 834
"498 200

"522 844
"547 882

"573 367
"599 278
"625 438
651 789
678 227
"704 638
"731 005
757 308
"783 552
"809 833

"836 143
"862 469

‘888 858

"915 399
941 825

968 407

995 080
"021 841
“048 686
"075 624

"102 664

"129 808

"157 016
"184 277
"211 592
"238 974

"266 487
"204 221

"322 247
*3.50 620

"023 023 |5°980 0631974 110
"954 1731973 958
928 1311973 817

"024 187
"024 366! °
"024 644| °
025 038
"025 485
"O25 Q11
7026 160] °
"026 351
026 438
7026 411

"026 367
"026 303
"026 244] °
"026 281
"026 310] *

"026 326] °
"026 389
"026 451
"026 516] °
026 582

"026 673
026 761
026 845
7026 938
"027 0.40

"027 144| °
"027 208 [4°97
"027 261
"027 315
027 382

027 513
°027 734
"028 026] °
028 373
"028 807

7] bq
0 OO GON An SOD = OC

wo

©o Go .
OO An HOw HO LO Or Qh + OW

> "
©

PROOKW HO WO OXI AN SP @ HW

w*

97

HF,
5 PER-CENT.
Logarithms and Co-logarithms of Dz, N,, and M,; with
their Differences.

Log M, A Colog M, A x

EE | TY | RN NS SRE fate ES | NSN ace

7994 23 | 025 724 14°009 028 | 992 12015990 972
6°O19 937 | 025 8go| ‘oor 1481989 750] -998 852
"045 827 | 026 042 13°990 898 | 987 837 |4°009 102

O
\°]
~y
co
ie)
©)
_
.@)

010250
O12 163

> fa wo Qo
WO Or An BW Hm O WO CATA BR YD MO Oo or ak ad & BO =

wr
Oo

hh GO YO me

"071 869
"098 052
"124 372
"150 827
"177 420
"204 160
"231 062

"258 143

285 412

*312 868
"340 510
*368 322

"396 290
"424 394
"452 628
"480 990
"509 481

538 114

"566 go2
"595 861
"625 007
"654 353
"683 912
"713 700
"743 73°
"774017
"B04 578
"835 430
"866 592
898 083
"929 925
"962 143

"994 762

5°027 811

"O65 325
"095 344
“129915
°165 087
"200 OII
"237 437

"274718
°312 806

026 183
026 320

026 455

026 593
026 740

026 go2
027 081
027 269
027 456
027 642
027 812
027 968

028 104
028 234
028 362
028 491
028 633

028 788
028 959
029 146
029 346
029 559
029 788
030 030
030 287
030 561
030 852
O31 162
O31 491
031 842
032218
032 619

033 049
033 514
034 019
934 57!
035172
035 824
036 526
037 281
038 088
038948

978 735
"965 140

"95° 593
"935 404
"920 183
"905 180
"890 857

877 283
"863 672
"849 588
"834 619
818 294
"800 412
"781073
"761 013
"740 462
"719 842
"699 611
679 846
"660 625
641 948
623 461
"605 207
"587 235
"569 398
"5651 701
"534 148

"516745
"499 422
"482 199
"465 100
"448 ITI

"431216
"414 420
"397 848
"581 543
"305 508

"349 717
"334025
"318 220
"302 156
"285 737

986 405
985 453

984 871
984 719
984 997
985 677
986 426

986 389
985 916
985 031
983 675
982 118

980 661

979 940
979 449
979 380
979 799

980 235
980 779
981 323
981 513
981 746
982 028
982 163
982 303
982 447
982 597
982 677
982 777
982 9o1
983 OL!
983 105

983 204
983 428
983 695
983 965
984 209

984 308
984 195
933 936
983 581
983 060

"021 265
"034 860

"049 407
"064 536
"079 817
"094. 820
"109 143
"122717
"136 328
"150412
"165 381
°181 706

"199 588
"218 927
238 987
"259 538
280 158

"300 389
"320 154
"339 375
*358 052
"379 539

"394 793
"$12 705
"430 602
"$48 299
"405 852
"483 255
"500 578
"517 801
"534 900
"5.51 889

"508 784
"585 580
"602 152
"618 457
"634 492

650 283
"665 975
681 780
‘697 844
"714 263

O13 595
O14 547
O15 129
O15 281
O15 003
O14 323
O13 574
013 O11
o14 084
o14 969
016 325
o17 882

O19 339
020 060

020 551
020 620
020 231

O19 765
O19 221
018 677
018 487
018 254

017972
017 837
O17 697
O17 553
O17 403
O17 323
O17 223
O17 099
016989
016 895

016 796
016 572
016 305
016.035
O15 791
O15 692
O15 805
016 064
O16 419
016 940

O

bq

aN wo &o is) iS)
0 OO CONIA £WOH HO CO CON An $& BW Hm O oO Cnr Qin OW WD He

© COArF Ar fh GO WD

ir
Oo

BP & Lb

98

HF.
5 PER-CENT.

Logarithms and Co-logarithms of D,, N
ther Differences.

2, and M,; with

O pw wo

~
PwWbe CO © CONT OA

~“

5
6
7
8
9

foe) co
OO COrx1 Air SO WH = O

ve)
.e)

. 2)
OO Or~37 Qn & & HD Hm

591 377
"561 824
"531 910
"501 618

"470 857
"439 259
"406 687
"372959
"337 928

"301 487
"263 754
"224 750
"184 296
"142 208

"098 289
"052 236
"003 274
2°959 954
894 929
"834 802
"779 331
“701 428
"629 744
"555 471
"479 090
"401 501
"323 O24
"240 506
"151 150

"052 790
"O42 419
817 681
"679 037
"528 122

"368 735
"207 860
"050 585
O'914 756
"807 546

"697 134
°574 353
424.645
1224 479
I°Q02 259

"979 447
"970 086
"969 708
"969 239
968 402
"967 428
"966 272
"964 969
"963 559
"962 267
"961 002
"959 540

"957 912
"956 081

953 947
"951 038
"947 680) _
"043 97513049 046
"939 873

"935 529
"931 097
"928 316
"925 727
"923 619

"922 411
"921 523
917 482
"910 644
"QO1 640
"889 629
875 262/2'057 581
"B61 356
"849 085
"840613

"839 125

"842 725
"864 171

"889 588
‘877 219
850 292

"799 834
‘677 780

A
7 aa tyr)

37 a4 3
"438 176
*468 ogo
"498 382

"529 143
"560 741
"$93 313
"627 O41
"662 072

"698 513
"736 246
"775 244
"B15 704
"857 792

‘Qo1 71I
"O47 704
"996 726

"105 O71
"165 198
"229 669
‘298 572
"370 256
"444 529
"520910
598 499
*676 976
"759 494
848 850

"947 210

"382 319
"320 963
"471 878
631265
"792 140
_ 949 415
"Bg2 790/T'085 244
"192 454
*302 866
"425 647
"575 355
775 52
0°097 741

"029 553
"029 O14
"030 292
"030 761

031 598
032 572
"033 728
035 031
"036441

"037 733
038 998
"040 460
"042 088

046 053
048 962
"052 320
"056 025
"060 127

064 471
"068 903
"071 684
"074 273
076 381

077 589
"078 477
"082 518
"089 356
098 360
"110371
"124 738
"138 644
"159915
"159 387
"160 875
"157 275
"135 82g
"107 210
"110 412

“149 708
"200 166
"322 220}1'Q02 259

608 3711959 119
"567 49091958 022
525 5121956 818
"482 330955 499

"437 8291954 083
"391 9121952 569
"344 4811950 960
"295 4411949 255
"244 6961047 448

"192 1441045 505
"137 6491943 391
"081 Of 01941 094
"022 1341938 595
"043 919 /3°960 729193 5 878

‘896 6071932 931
829 5381929 834
"759 3721926 627
"685 9991923 357
"609 356/920 095

"529 451/916 O11
"4-46 3621913 859
"360 2211910 622
"270 8431907 096
"177 9391903 959
“080 998/898 082
2°979 080/891 665
870 745/884 062
754 807/875 593
630 400/866 595

"496 995/858 004
"354.999|851 143
"206 142/846 896
"053 038/845 968
"899 006/848 229

‘747 235/851 980
599 2151855 683
454 898/852 291
"307 1891834 747
“141 9361806 799
"122 781 |0°948 735/761 705
"710 4401683 181
"393 621|508 638

an
© cn? Aun

Ff GW HN eH 3S

~~ ~~ OC
OW Or~7 Qh F&O HW ew O OH DHr~37 An PO HY HF CO WO D~) Qn

CO

co

‘Oo

© wr ak O&O YW mH OO.

Logarithms and Co-logarithms of D,, N,, and M,; with

z

55
6

7
8

Colog N,

SS

5°35! 754

"391 629
"432 510
"474 488

9 | °517 670

60

Ov
wm b&w wb -»

“I ~
0 OO ONAN POH HO ©) CNOA

Lo)

fe «)
O O71 Qh Po DH

go
I
2

3
4

"662 171

608 088
"055 519
"794 559
"755 3O4
807 856
"862 351
"918 Q60
"977 866

4039 271

"103 393
°170 462
"240 628
*314 007
"390 644

"479 549
"553 938
"639 779
°729 157
"822061

"91g 002

3'020 920

"129 255
"245 193
369 600

*503 005
"64.5 OO1
"793 858
"946 962

2°100 994

"252 705
"400 785
"545 102
"692 811
858 064

95 |T'051 265

6
7

289 560
"606 379

8 jo°097 741

9

A

Ce aaa Kcememmmmmmmereemmnemmemm ence] (Wem

99

HF.
5 PER-CENT.

their Differences.

Log M,

039 875 |3°268 797

040 881
041 978
043 182
044 501

045917
047 431
O49 040
050 745
052 552
054 495
056 609
058 906
O61 405
064 122

067 069
070 166
073 373
076 643
©79 995
083 089
086 I41
089 378
092 904
096941
101 918
108 335
115938
124 407
133 405

141 996
148 857
153 104
154032
15177!
148 020
144 317
147 799
165 253
193 201

238 295
316 819
491 362

"251471
"233 863
"215 997
"197 867

"179 330
"159 729
°138 870
"116.495
"092 421

"066 544

"039 143
"O10 335

2°979 880

947 596
"913 260
876 508
"836 302
"792 103
"743 524
"690 104
631 588
"567 906
"507 310
"432 117
*361 029
289 313

"217 S10

"143 831
059 003

1°966 578

861 106
739 837
"603 171
"452 672
"292 261
"129 OI!

0°970 485

"836 356
*736 730
635 348
"522 304

"382 752
"192 824

1T’881 070

A

Colog M,

982 674 |4°731 203

982 392
982 134
g81 870
981 463

980 399

979 141

977 625
975926
974 123

972 599
971 192
969 545

"748 529
"766 137
"784 003
"802 133
"820670
"840271
"861 130
883 505
"997 579
"933 456

"960 857
"989 665

967 716 |3°020 120

965 664

963 248
959 794
955 801
951 421
946 580

941 484
936 318
933 404
930 807
928 g12
928 284
928 197
924 321
917 172
997 575
894 528
878 731
863 334
849 501
839 589
837 350
840 874
865 871
900 374
898 618

887 046] -

860 358
810072
688 246

"052 404
"086 7.40
"123 492
"163 698
‘207 897
"256 476
"309 896
°368 412
"4.32 004
*498 690
"567 883
638 971
°710 687
"782 490
"858 169
"949 997

2°033 422

138 894
"260 163
"396 829
"547 328

"797 739
"870 389

T'029 515

"163 644
"263 270

A

O17 326
017 608
017 866
018 130
o18 537

019 601
020 859
022 375
024.074
025 877
027 401
028 808
030455
032 284
034 336
036 752
040 206
044 199
048 579
053 420
058 516
063 682
066 596
069 193
o71 088
O71 716
071 803
075 679
082 828
092 425
105 472
121 269
136 666
150 499
160 411
162 650
159 126
134 129
099 626
101 382

112954
139 642
189 928
311 754

WN
# G BY = O 6 wrt an

“A
ww

~“ ~
nr FO H HO OO CON A

Cc

Oo
O ONT AM BO HHO 6 CYA

© ww awk hop d

we bee m=
Oo COr~ An PW D&O

$f DW YN KK O

bo

so

Oo

5
6
7
8
9
°
I
2
3
4
5
6
7
8
9

>
OO Hb = O

aN

mBROoO Hm CO DO C™ AN

16°507 5
16° 387 4
16°279 6
16°181 9
I6°0QI 5
16°005 6
15°931 5
15°836 3
15°74 5
15°648 7

15°5419
15°432 0
15°322 6
15°217 2
15°120 3

150341
14°958 4
14°887 1
14°818 3
14°748 4

14°673 2
14°591 6
14°502 7
14°406 I
14°304 7
14°198 1
14°085 4
13°967 8
13°845 0
13°7166

13°582 4
13°442 8
13°297 I
13°145 0
12'986 2
12°8204
12°647 3
12°465 0
12°272 5
12°069 2

11°854 6
11'6299
11°3968
LI'156 5
10°909 8

100
HF.

5 PERCENT.

Assurance of a Unit.

A;

166 311 }°
1720271"
177 161 |"
181 816}:
186 119|°
"190 212]°
"194 212]°
‘198 270°
"202 547 |"
"207 204 |°
"212 2901"
"217 524)"
"222 7351)"

"227 753)"
"232 368 |-

"230471|°
"240077 |°
243 473 |"
"246 746 |°
"250 074)"

"253 658°
"257 543 |"
261 778|°
"266 379°
"271 202]°
"2.76 280]:
"281 646]:
"287 2471"
"293 096°
"299 208 | °

"395 599 | °
"312 250|°
"319 183)"
"326 428]°
"333 991 |"

"341 885|°
"359 129)"
"358 B10)"
"367977"
377 659)"
"387 875)°
"398 576°
“409 676 |°
"421 118)"
"432 869 |°

55
6

7
8

“
Oo

QD pweon

~J
OO cox~r Qn BOK wNO WO COA

‘© co co “I
POwobm O OO O~r Qn SOW * O

\O
© cnr Qn

~

10°6579

10°399 0
IO°I31 4
9°8537
9°565 6
9°267 7
8°967 I
8°665 5
8°365 3
8°068 o

777142

"444 861
"457 188
‘469 936
"483 155
"496 877
"STI 062
525 375
"539 738

"554 933
"568 189

*582 180
"596 196
"610 350
624 574
"638 834

"653 087
667 225
680 813
"693 664
"705 659

"716 643
"726 535
"735 324
743 989
"752 741
“761 972
772 345
*784 307
796755
-808 822

819 952
"829 252
"835 904
"839 719
"840 524

"838 545
"835 125
"831 574
"834 834
"B49 540
867 388
887 238
"908 045
929 706
"952 381

Values of Annuities, and Single and Annual Premiums for

7038 159
"040 108
"042 217
044 515
7047 028

"049 774
"052 711
7055 842
7059 158
"062 658
"066 351
"070 307
"074 59!
"079 221
"084 229
"089 646
"095 478
"IOI 570
"107 828
"114.163

"120 434
°126 513
"132 295
"138 385
"144 969

"152 437
"161 553
"173 153
"186 675
"201 463

*216 860
"231 266
"242 571
"249 476
"25° 977
"247 318
"24.1 200
"235 110
"240 690
268 869
*311 460
"374 678
"470 232
629 800
"952 381

HF.
SIX. PER-CENT.

ww _ b=

we

>

Go
Oo COMP Or. PO Ym EO WO CONAN POH HO OO ON AN BPO YW HO

>
ro]

Re wp

a
mown CO OO CON? OA

102

Hr.
6 PER-CENT.
Commutation Table.

ee: | oS | Seat Fg 9 OPS

55 839°5 |79985t°0 111345 587 | 7404717] 165 052°1
52 513°3 (74733777 119545736 |7238°76) 157 648'0
49 332°2 |698005°5 | 9798399 | 7 030°03| 150 409°2
46 300°7 651 704°8 | 9100393 |6790°92| 143 379°2
43 422°9 |608281°9 | 8448688 | 6 533°941 136 588°3

40 701°3 1567 580°7 | 7840407 |6270°23 | 130054°3
38 134°9 [5294458 | 7272826 |6007°67/123 7841
35722°7 1493 723°1 | 6743380 |5754°03/ 117 776'4
33 462°4 [4602607 | 6249657 |5515°80] 112 022°4
31 350°5 14289103 | 5789396 | 5 297°99] 106 506°6

29 377°9 1399 532°3 | 5360486 | 5 099°99 | 101 208°6

27 524°4 13720079 | 4960954 |4909°38| 96 108°6o
25777°% 13462307 | 4588946 |4720°12| 91 199°22
24125°4 |322105°3 | 4242715 1452744] 86479°10
22 559°0 1299 546°3 | 3920610 |4326°64| 81951°66
21072°6 1278 473°7 | 3621063 |4117°18| 77 625°02
19 664°9 |258808°8 | 3 342590 |3902°20] 73 507°84
18 340°7 1240 468'2 | 3083781 |3 691710] 69 605°64
170981 [223 370°1 | 2843313 |3 486°67| 65 914°54
15936°5 1207 433°6 | 2619943 |3292°88| 62 427°87

14.854°7 11925789 | 2412509 |3113'20|] 59134°98
13 847°8 [17873171 | 2219930 |2947°14| 56021°78
I2QII'1 |165820°0 | 2041199 |2794°20! 53074°64
12 039°3 [153 780°7 | 1875379 |2653°28| 50280'44
11225°5 11425552 | 1721598 |2520°88} 47 627°16

10 465°9 | 132 089°3 | 1579043 |2396°76| 45 106°28
975740] 1223309 | 1446954 |2280°65) 42 700°52
9995'551113 2364 | 1324622 [2171'1I| 40428°88
8 477°37|104759°0 | 1211385 |2067°77| 38257°77
7900°03| 96858'96; 1106626 |1970°28| 36 Igo°00

7 360°89| 89 498°08| 1009768 |1878:30] 34219°73
“6857°10] 82640°98] 920269'4)1 79117] 32 341°42
6 386°50| 76254°48 837 628°5|1 70871) 30550°25
5947°05| 79307°43| 761 374°0|1630°75| 28 841°54
5 536°64| 64770°79| 691 066°6|1 556'98| 27 210°79

5 153°36| 59617°43] 626 295°8)1 487°09) 25 653.81
4795°45| 54821°97| 566678°3|1420°88| 24 166°72
4461°75| 50360°23} 511 8564/1 358°62| 22 745°84
4150°76| 46209°47| 461 4961/1 300°18| 21 387°22
3860°97| 42348°50| 415 286°7| 1 245°34] 20087°04

3.59085] 3875765} 3729382/1193°76| 18841°70
3338°63| 35419°02| 334180°5/1144°80| 17 647°94
3102°55| 32316°47| 298 761°5|1097°69| 16 503°14
2881-21] 2943526} 266445°0/1051'98| 15 405°44
2 673°53| 26761°73| 237 009°8|)1007°38| 14 353°47

we bw ee]
© cor~T Qn + & Ym OC

)
O wr7 An B&H HW WO OO CHAT An BPW WHO

Go

&o

7 pwenyS

oy
PhO wwe O KO CONT AN

ve] co oO ~r ~“ “A A wn
0 OO O™ ANN PRON H CO OO CON ON #&O HMO W/O Wr~? AN BWW HO CO ON QNn

\o
YO Cr-tr Qn SP HW

2 478°35
2 295°35
2124°12
1 964°04
1 814°43
1074°41
I 542°23
I 417°30
I 299°02
1 187°05

1081°21
981°885
889°090
802°360
721°381

645°847
575° 387
509° 192
447°141
389°316

335°783

286°729
242°355
203°541
169°927

141°177

HF.

6 PER-CENT.

3 268°283
2 759°09I
2 311'950
I 922°634
I 586°851
I 300°122
1057°767
854°226
684°299
§43°122
426°156
329°447
250°228
186°349

13 5°897
97°136
68°326
47°587
33°975
23°115
16°303
11°606

8°203
5°59
3° 545
2°035
975
"312
"000

8,

210 248°1
185 964°7
163 976°6
144 112°7
126 212°8

110 127°4

95 716°37
82 847°56
71 396°06
Ol 243°57

52 278°13
44 393°89
37 491°55
31 478°29
26 267°39
21 777°88
17 934721
14 665°93
11 906°84

9 594°885
7672°25!
6 085°400
4 785°278
3727512
2 873°285
2 188°986
1 645°864
I 219°708

890°261

640°033

453°O84
317°787
220°651
152°325
104°738
71°663
48°548
32°245
20°639
12°436
6°867
3°322
1°287
“312
"000

Commutation Table—(continued).

Mz

963°530

920°819
879°515
839°664
801°233

763916
726° 509
688°872
650°821
612°378

573°732
53.5608
498°391
461°987
426°424

391°723
357°821
324°195
290°966
258°451

226°955
196'G07
168°763
143°667
121°575

102°443
86°223 2
72° 586 7
60°571 3
49°7152
39°904 4
31°068 7
23°3115
16°871 1
11819 I
8-087 5
5°53 3
3°7746
2°746 5
2°109 5
1°708 I
I°3100
"9449
“607 I
"294 7

R,

13 346°08

12 382°55
11 461°73

10 582°22
9 742-554

8 941°322
8 177°406
7 450°897
6 762°025
6 I11°204

5 498°826
4925094
4 389°486
3 891°095
3 429°108
3 002°684
2 610°961
2 253°140
1. 928°945
1 637°979

1 379°528
1 152°573
955°666
786°902
643°235
521661
419°217
332°994
260°407
199°836
150°12I
I10°217
79°148
55°836
38°965
27°146
19°059
13°555
9°781
7034
4°865
3°157
1°847
"902
"295

&
2 S coaoran

tr RPO DD &

“sr

CO ~7
On ROH mM O OO CD71 AN PO Pw CO YO CNT OA

co
Oo conn A

‘oO
oO

0 wow an f OF YD m

104

HF.
6 PERCENT.
arithms of D,, N,, and M,; with

Logarithms and Co-

» i] =
OO C~r Ain © WY HO

o

> _ &o & ,
b&w He O OO COW~3 Ain F&O bm OO OO CON QAM PW D&O

oy
POWOWKHH CO WO GO~? AN

Log D

490 270
693 130
665 587
637 719
"609 608
“581 322
552 944
"524.557
"496 244
"468 o21
"439 718
"411 235
"382.475
"353 320

"323 719
"293 O91
"263 415
"232 947
"202 392

A
* | (Log epz)

972 86

973.487 °
"972 132! °
971 889) °
"O71 714! *
"971 622) °
‘971 613} °
971 687) °
971777) *
"971 697| °
"O71 517°
"971 240} °
"970 845] °
"979 399) °
"969 972) °
"969 724) °
"969 5321 °
959 445} °
"969 473} °
"171 865] °
"141 382) °
"110 962) °
"080 602] °
"050 204] °

7019 778] °
3°989 334] °
958 829) -
928 261! -
"897 62g) °
"866 930] °
°836 140] °
"B05 263] °
°774 301) °
743 247| °
"712 091| °
"680 830] :
"O49 505] °
“618 128] -
"586 696] °

"555 197
"523 509
"491 718
"459 575
"427 085

968 372 °
968 149] °
"967 857] -
"967 510} °
"967 077) °

Colog D

eir Differences.

027 140
"027 543
"027 868
‘o28 I11

028 286
028 378
"028 387
028 313
"028 223

A
7 (Colog vpz) Log

7028 303
028 483
"028 760
"029 155
"029 601
"030 028
"030 276
"030 468
"030 555
"030 527
"030 483
"030 420
030 360
"030 398
"030 426

"050 444
030 505
030 568
030 632

"030 799
"030 877
"030 962
"031054
031 156
"031 261
"031 325
"031 377
"031 432
"031 499
031 628
031 851
"032 143
"032 490
"032 923

1873 §17970 342
"843 8591970 192
"B14 0511970 054
"784 105)9°9 923
"754.028
*723 822/969 661
"693 4831969 521
"663 004/969 363
"632 367/969 185
"601 552/969 oco
"570 5521968 814
"539 306/968 632
"507 9981958 466
476 4641968 320

444 7841968 195
"412 9791968 079
"381 0581967 967
"349 0251967 854).
"316 8791967 730

‘284 6091967 59!
"252 2901967 437
"219 6371967 265
°186 902/967 081
"153 983/966 885
"120 8681966 672
087 5401966 446
053 9861966 205
"020 191/965 949
"030 699 |4°986 1401965 674

"951 814/965 382
"917 196/965 069
"882 2651964 736
‘847 001/964 378
"811 3791963 994

775 3731903 582
738 9551963 133
"702 0881962 643
"664
"626 838 961 520
"588 3581960 879
"549 2371960 187

"509 4241059 444
"468 868/958 646

“427 5141957 795

969 794

731\962 107

> Go wo re) iS) - ad
0 OO G7 An FPO HK CO HO Or AN HOH eH CO YW GO~ Qh POW WH = O

bh O DH

p.

qn
OH Ye Oo WO Com Qn

105

HF.
6 PER-CENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

Log M, A Colog Mz

baq

10 |6°096 991 | 029 492 |13°869 476 | Qyo 188 |4°130 524 | 009 812

PO LY

bm

we

i]
© @O~ AN PON KH OC YW G7 AN FO HH QO DO O™ AN

eS) &o

tn
6 ox anh pwemnd

& © DN mm

"126 483
"156 141
"185 949
"215 895
"245 972
276178
“306 517
"336 996
"367 633

"398 448
"429 448
"460 634
"492 002
"523 536
"555 210
587 O21
618 942
"650975
°683 121

"715 391

"747 800
"780 363

"082 804
"117 735
"152 999
"188 621
"224 627
"261 045
"297 O12
"335 269
"373 162
"411 642
"450 763
"490 576
"531 132
"572 486

029 658
029 808
029 946
039 077
030 206
03° 339
030 479
030 637
030815
031 000
031 186
031 368
O31 534
031 680

031 805
031921
032 033
032 146
032 270

032 409
032 563
032 735
032 QIQ
033115

033 328
033 554
033 795
034051
034 326

034 618
034931
035 264
035 622
036 006

036 418
036 867
037 357
037 893
038 480
039 121
039 813
O40 556
O41! 354
O42 205

"859 064
"846 957
831929
"815.175
"797 284
"778 706
"759 972
‘741 608
"724 III

"797 57°
‘691 027
673 953
655 853
"636151
‘614 600
"591 310
"567 156
"542 411
"517 577

"4.93 207
"469 401
"446257
"423 782
"HOI 552
"379 625
"358058
*336 O81
"315 502
"204 527
"273 766
"253 137
"232 668
"212 388
"192 283

"172 337
"152 558
"133 097
"114 004
095 288

‘o76919
7058 731
"040 481
"022 006
"003 195

987 293
984 972
983 246
982 109
981 422
981 266
981 636
982 503
983 459
983 457
982 926
981 goo
g8o 298
978 449

974710
975 846
975 255
975 366
975 630
976 194
976 856
977 525
977770
978 073
978 433
978 623
978 821
979 025
979 239

979 37!
979 531
979 720
979 895
980 054
980 221
980 539
980 907
981 284
981 631

981 812
981 750
981 525
981 189
980 670

‘140 336
"153 943
*168 071
"184 825

"202 716
"221 294
"240 028
258 392
"275 889

"292 430
"308 973
"326 047
‘344 147
"363 849
"385 400
"408 690
"432 844
457 589
"482 423

"506 793
"530 599
"5.53 743
"576218
"598 448

"620 375
"641 942
"663 319
"684 498
"795 473
726 234
"746 863
"797 332
°787 612
"807 717
"827 663
847 442
"866 903
"885 996
"QO4 712
"923 081
‘941 269
"959 519
"977 994
"996 805

O12 707

015 028

016 754
o17 891
018 578
018 734
018 364

017 497
016 541

016 543
017 074
018 100
O1g 702
O21 551
023 290
024 154
024 745
024 834
024 37°
023 806
023 144
022 475
022 230
021 927
O21 567
O21 377
021 179

020 975

020 761

020 629
020 469
020 280
020 105

019 946
019779
O19 461
O19 093
018 716
018 369
018 188
018 250

018 475
018 811

O19 330

p > & &o w

ir

106

HF.
6 PER-CENT.
Logarithms and Co-logarithms of D,, N., and M,; with
their Differences—(continued).

A A .
Log D, (Log ep) Colog D, (Colog vp.) Log N,

YO conr Qt

“WN

OO Gr AN PON HNO OHO GO QwH PO HN HO

nN

~

“I

CO

Lo 6)
So nm O WO CO~?3 AN GW BO mM O

\O

Ve)
© GD An

i}
e eo e

e 1]

“300 849
"327 179
"293 150
°258 741
"223 863
"188 149
"151 460
"113 616
"074 468

"035 QIO0

639 151
"672 821
"706 850
"741.259
"776 137
811 851
"848 540
"886 384
"925 532
"966 ogo

"956 885/3°007 939
955 424) "O51 054
953 794 °095 630
"951 905) °141 836

"949 831) *189 871
"946 921] "240 040
943 563} °293 119
939 859] 349 556
"935 759 °409 697
"931 413) °473 941
926 980, +542 528
"924 199] “615 548
"921 611} “691 349
919 503) °769 738

"918 294] "850 235
"917 407] “931941
"913 365|2°014 534
"906 527) "ror 169
"897 524) "194 642
885 513] ‘297 118
"871 144] *411 605
"857 240} “540 461
"844 969] °683 221
"836 496) °838 252
*835 oo8l1-o01 756
*838 608] -166 748
"860 056] *328 140
"888 673] °468 084
885 471| “579 411
873 103) "693 940
846175) °820837
"795 717| °974 662
"673 664)0°178 O45

3°394 162/1°966 687/4'605 838) 0°033 313 |4°385 309] 956 878
"966 330
905 971
"995 59!
"965 122
964 286
963 311
"962 156
"960 852
"959 442

"958 151

"033 670
"034 029
"034 400
034 878

"035 714
"036 689
"037 844
"039 148
"040 $58 |3°952 5721944 187

"041 849
"043 115
"044 570
"046 206
048 035
"050 169
"053 979
'056 437
"060 I4I
"O04 244
068 587
"073 020
075 801
"078 389
‘080 497
‘081 706
"082 593
"086 635

"093 473
“102 476

"114 487
"128 856
"142 760
"155031
"163 504
"164.992
"161 392
"139 944
"III 327
"114 529
"126 897
"153 825 | *308 518) 680 356
"204 283 |1°988 874] 505 845
"326 336] 494719

"342 187/955 878
"298 065/954 785
"252 859/953 583
206 433/952 262

"158 6951950 843
"10g 538/949 325
"058 863/947 710
"006 573/945 999

"896 759) 942 238
"838 997/940 113
‘779 110/937 802
"716 912/935 288
"652 200/932 546

"584 746) 929 574
"514 320] 926 446
"440 766| 923 212
"363 978/919 919
"283 897| 916 639

"200 5361913 448
"113 984/910 406
024 390] 907 183
2°931 573/903 73
"83.5 246] 899 651
"734 897) 894 672
"629 569) 888 217
"517 786] 880 550
"398 336] 871 Oo!
"250 327| 862 882

"133 209} 854 170
1°987 379| 847 207
"834 586] 842 906
"677 492/842 004
"519 496} 844 399
"363 895) 848 384
"212 279/852 403
"064 682] 849 268
0°913 950| 831 806
"745 756) 803 913

"549 669) 758 849

WO Or~7 Qu

CO ~r ~ “A A
O OI AX pwn id O COr~1 An BW H HE O OO GD Air RBWH& OC

Ve)
re)

\o
OO ON AN APH HD =

Logarithms and Co-logarithms of D,, N,, and M,; with

z

Colog N,

55 |5°O14 O91
6

\O
oO

Ve)
WO GOT Qtr HO wv =

657 813
"7901 935
"747 150
"793 567
"841 305
"890 462
"941 137
"993 427

4°047 428

"103 241
‘161 003
‘220 890
°283 088
"347 800
"415 254
"485 680
"559 234
"636 022
"716 103

"799 40.4
"886 016
‘975 O10

3°068 427

"104 754
265 103
"379 431
"482 214
"601 664
"729 673
866 791

2°012 621

"105 414
"322 508
“480 504
636 105
"787 721
"935 318

1°086 o50

254 244

"459 331
691 482

O'OL! 126

"505 285

107

HF.
G6 PERCENT.

their Dafferencese—(continued).

241 151
319 644

Lan!

Log M,

043 122 |2°983 865

"964 174
"944 244
°924 106
"993 759
"883 045
“861 241
"838 139
"813 461
"787 020

"758 799
"728 847
"697 570
"664 630
629 842

"592 979
"553 965
510 806
"463 842
“412 379

"355 949

"294 262
"227 278
"157 357
‘084 843
"010 484
"935 624
"860 857
‘782 267
696 489
"601 020
"492 323
"367 570
"227 143
"072 583

0907 814

740 619
$76 871

232 511

“117 259

494.155 (1'975 387

“783 281
"409 414

A

980 309 [3016 135

980070
979 862

979 953
979 286

978 196
976 898
975 322
973 559
971 689

970 138
968 723
967 060
965 212
963 137
960 686
957 141
953 036
948 537
943 561

938 322
933 016
939 979
927 486
925 641

925 140
925 233
921 410
O14 222
904 531

891 303
875 247
859 573
845 440
835 231
832 805
836252
861 905
897 592
896 143
884 748
858 128

Colog M;

2

035 826
"055 756
075 894
"096 241

"116.955
138.759
"161 861
"186 539
"212 980

"241 291
"271 153
"302 430
"335 37°
"370 158
"407 O21
"446 335
"489 194
"536 158
"587 621
644 060
"705 738
"772 722
‘842 643
915.157
"989 516
"064 376
"E39 143
"217 733
°303 511

"398 980
"507 977
"632 430
"772857
"927 417
"og2 186
"259 381
"423 129
“561 224
663 632

"767 489

"882 741

807 894 |0°024 613

686 133

"216 719
"530 586

A

O19 691

019930
020 138

020 347
020714
021 804
023 102
024 678
026 441
028 311

029 862
031277
032940
034 788
036 863

039 314
042 859
046 964
O51 463
056 439
061 678
066 984
069 921
O72 514
974 359
074 860
074 767
078 590
085 778
095 469
108 697
124753
140 427
154 560
164 769
167 195
163 748
138 095
102 408
103 857
115 252
141 872
192 106
313 867

&

wn
OO On? Qn

ve) Ve) fos) (oe) ~ ~Y A o,)
© G7 An FON HO OO GO An #$O Ym OO CO Wr 3 Qn HONK C OO WO AN BW YP HO

|
|

& w (] = =
7 6 ‘© COMI AC. wm O WO CON At O&O Be O LO CO~N AN FW HW HO

tn
hw HO = CO OO DOT Qn SO HD =

az

14°324 I

14'231 4
I4'I4Q 1

14°008 3

13°945 0
13°883 5
13°'8210
13°754 6
13°6811

13°599 7
13°5156
13°4317
13°3513
13'278 3
13°215 0
13°I161 0
13°111 2
13°0064 I
13°016 3

12°964 I
12°906 8
12°843 3
12°773 2
12°699 3
12°6209
12°537 3
12°449 6
12°357 5
12°260 6

12°158 6
I2°O5IYg
11°940 0
11°822 2
11°698 6
11°568 7
11'$32 1
11°287 1
11°132 8
10°968 4

10°793 5
10°608 g
10°416 I
10°216 3
10°009 9

As

"132 597
"137 846
"142 504
14°075 51°
"150472

"154055
"157 537
"161075
"164 836
"168 og2

"173 599
"178 365
"183 112
°187 663
"IOL 792

"195 380
"198 435
"201 253
203 922
"206 626

"209 577
"212 823

"216 419
"220 384
"224 568

"229 006
"233 735
°238 700
"243 916
"249 401
"255 174
"261 214
267 550
"274.212
‘28r214

"288 567
296 298
"304 503
"313 239
“322 546

"332446
"342 896
"353 804
"365 116
"376 799

146 670

108

HF

6 PER.CENT.
Values of Annuities, and Single and Annual Premiums for

Assurance of a Unit.

008 653
"009 050
"009 407
"009 729
"010 026
"010 308
"010 585
‘010 868
‘O11.172
"OIL 511

“ort 891
"012 288
"012 688
"013 076
O13 432

"O13 745
‘O14 013
"014 262
‘O14 500
"O14 742
"015 008
"O15 304
"O15 634
"016 001
"016 393

016 813
"017 266
‘017 748
018 261
"018 808

‘O19 392
"020 014
"020 676
"021 386
"022 145

022 959
'023 833
"024 782
"025 818
"026.950
028 189
"020 537
"030 992
"032 552
"034 224

Zz

~ ~“ vn n in
8 \o CONF AG bs be o © wr AN Pwo be oO OO CONTA BOK KH O O Or Qn

Oo
© OTF Qwm OO DN mo

ay

A;

°388 780

"4OT 167
"414 061
"427 520
"441 589

"456 229
"471078
"486 047
"501 009
"515 884

"530 639
"545 489
"560 563
"575 785
"591 123
"606 527
621 879
636 685
"650 726
"663 859

675 898
"686 737
"696 347
"705 840
"715 452

‘725 637
"737 165
759 571
"764 603
"778.275
"790 931
"B01 547
"809 154
"813 512
814 395

"812 027
"807 918
"B03 545
"806 976
"823 720

"844 210
"867 154
‘891 352
"916 697
"943 397

"036 004

"037 920
“O40 000
"042 271
"044 762

‘047 491
"050 413
"053 53°
"056 833
"060 318

"063 904
'067 954
"072 206
076 828
"081 833

"087 253
"093 093
"O99 194
"105 457
"111 789
*118 044
"124 087
"129 806
"135 821
"142 321

"149 706
"158755
"179 329
*183 857
198 679
"214138
"228 620
‘239 989
"246 921
248 365
"244 522
238 081
"231 522
"236 643
"264 496

"306 729
‘369 479
"404 375
622 884
"945 397

IT.

H™(5),

HEALTHY MALE LIVES,

OMITTING THE

FIRST FIVE YEARS OF ASSURANCE,

I
2
3
4
5
6
7
8
9
30
I
2
3
4
35
6
7
8
9

on pweKys

iw
PON me O © CNT AN

110
H™(),

Elementary Values.

Log [,

4°000 000
3°998 259
"996 774
"995 460
"994 185

"992 863
"991 448
989 806
"987 890
"985 516
"982 588
"978 956
"974 742
"970 254
"965 578
"960 851
"956 265
"951 872
"947 532
"943 297

"939 170
"935 154
"932 153
"927 114
"923 089
918 973
"914 608
"QI0 O91
"905 418
"g00 586

"895 699
"890 756
"885 813
"880 756
"875 582
"870 228
"864 570
"858 597
"852 297
"845 594

838 471
"830973
823 083
"814 847
"806 112

Log pz

1998 259

"998 515
“998 686
"998 725
"998 678
"998 585
"998 358

“998 084

"997 626
"997 972
"996 368
"995 786
"995 512
"995 324
"995 273

"995414
"995 607
095 660
995 795
"995 873

"995 984
"995 999
"995 961
"995 975
"995 884

"995 935
"995 483
995 327
995 168
995 I 13

995 957
995 957
"994 943
"994 826
994 646

"994 342
"994.027
"993 700
"993 297
"992 877

"992 502
"992 I1O
"991 704
"QQ 265
"990 809

000 256
000 171
000 039

999 953
999 997

999 773
999 726
999 542
999 446
999'296

999 418
999 726
999 812
999 949

000 141

000 193
000 053
000 105
000 108
000 [II

O00 OT5
999 962

000 014
999 999
999 75!

999 848
999 844
999 841
999 945
999 944

000 000

999 886
999 883
999 820
999 696

999 685
999 673
999 597
999 580
999 625

999 608
999 654
999 501
999 544
999 447

Colog 2,

4°000 000

"OOr 741
"003 226
"Od 540
7005 815

"007 137
“008 552

"O10 194

"OIZ2 110
"014 484

"O17 412
"O21 O44
025 258
"029 746
"034 422

"039 149
"043 735
"048 128
"052 468
7056 703
"060 830
"064 846
068 847
072 886
"076911
“081 027
"085 392
"089 909
"094 582
"099 414
"104 301
"109 244
“114 187
"119 244
"124 418

"129 772
"135 430
"141 403
"147 793
"154 406
"161 529
"169 027
"176.917
"185 153
"193 888

a

= bm
0 OO GOI AN PRO YP & O

wo

~]

Go &o
OW GN AN FPOHKHK O OO GAH HO WD

+. >
°

&O WD mw O LO COr~? ACtr PD

tA

bong

ae

wo

&o

&o

°
I
2
3
4
5
6
7
8
9
fe)
I
2
3
4
25
6
/
8
9
°
I
2
3
4
5
6
7
8
9

Bw -N
am PO YN mm O

wr
POD HO OO DOr An

O°001 741
“oor 485
“OOI 314
"OO1 275
"OOL 322

"OO 415
"001 642
"001 916
"002 374
"002 928
"003 632
"004 214
"004 488
"004 676
"004 727
"004 586
"004 393
"004 340
004 235
"004 127
"00.4 036
"O04 OO!
"O04 039
"00.4025
"004 116

“004 365
"004 517
"004 673
"004 832
"004 887

"004 943
"004 943
"005 957
"005 174
"005 354
"005 658
"005 973
"006 300
"006 703
"007 123

"007 498
"007 890
"008 236
"008 735
"009 191

999 744
999 829

999 961
000 O47

000 093

000 227
000 274
000 458

000 554
000 704

000 582
000 274
ooo 188
000 O51

999 859

999 807
999 947
999 895
999 892
999 889

999 985
000 038
999 986
000 O°9!
000 249

000 152
000 156
000 159
000 055
000 056

000 000
000 114
000 II7
000 180
000 304

000 315
000 327
000 403
000 420
000 375

000 392
000 346
000 499
000 456
000 553

Log d,

1°602 060

"531 479
“477 T20
"462 398
"477 121
"505 150
568 202
"633 468
"724 276
"812913

*903 090
"963 788
‘986 772

2°000 000

“O00 000

1°982 271

"959 O41
"949 390
"934 498
*919 078

"903 090
"897 627
897 627
"Bg2 095
"897 627

‘919 078
"929 419
939 519
"949 390
"949 390

"949 390
"944 483
"949 390
"954 243
"963 788

"982 271

2°000 000

"017 033
"037 426
"056 905
071 882
"086 360
"096 910
"113 943
"127 105

A

929 419

945 642
985 277
O14 723
028 029
063 052
065 266
ogo 808
088 637
090 177
060 698
022 984
013 228
000 000
982 271

976770
99° 349
g85 108
984 580
984 o12

994 537
O00 000
994 468
005 532
O21 451

O10 341
O10 100
009 871
000 000
000 000

995 993
004 907
004 853
009 545
018 483

O17 729
017 033
020 393
019 479
014.977

014 478
O10 550
017 033
O13 162
O15 Qto

070 581
054 358
O14 723
985 277
971971

936 948
934 734
909 192
QII 363
909 823

939 302
977016
986 772
000 000
O17 729
023 230
009 651
014 892
O15 420
o15 988

005 463
000 000
005 532
994 468
978 549

989 659
989 goo
99o I 29
O00 9000
000 000

004 907
995 993
995 147
99° 455
981 517
982 271
982 967
979 607
980 521
985 023
985 522
989 450
982 967
986 838
984 ogo

09 & ve w oe] ong
Oo COnTAN $O HHO OO O87 An LPHH HO LO CHxr Air AW WO HO

tA
© Or ah pw pans

mi wb O

ir
\O corr Aun

NA
oO

>)

“I

I
2
3
4
5
6
7
8
9

70
I
2
3
4
5
6
7
8
9

\O co co
C CO @W71 An PW HW & OC

ak — & YN &

4

Elementary Values—(continued).

Log ¢,

3°79 921

°787 177
*776 846
765 818
"754 119

"741 O24
728 110
“713 575
"697 839
"680 789
662 286
"642 465
621176
"598 462
"574 147

"548 144
"519 959
"489 114
"454 997
“417 139
"375115
"329 Oo1
"279 895
*226 600
"168 792
"105 851
037 426
"962 369
"880 242
‘790 988

"694 605
"592 377
"484 300
369 216
"247973
"117 271
973 128
"806 180
‘612 784
°361 728

"O41 393

0°602 060

112
HX(5),

Log pz

T'990 256

"989 669
"988 972
988 301
"987 505
"986 486
"985 465
"984 264
982 g50

"981 497,

*980 179
978711
"977 286
"975 685
"973 997

971815
999 155
"965 883
"962 142
"957 976

"954 486
"95° 204
"946 705
"O42 192
"937 959
"931575
"924 943
"917 873
"910 746
"903 617
‘897 572
892 123
884 916
878 757
"869 298

"855 857

833.052
"806 604
"748 944
"679 065

"560 667

A

999 413

999 303
999 329
999 204
998 981

998 979
998 799
998 686
998 547
998 682

998 532
998 575
998 399
998 312
997 818

997 340
996 728
996 259
995 834
996 510

995 808
996 411
995 487
994 867
994 516

993 368
992 930
992 873
992 871
993 955

994 551
992 793
993 841
990 541
986 559

977 195
973 552
942 340
930 721
881 002

Colog ¢,

4°203 079

"212 823
"223 154
"234 182
"245 881
258 376
"271 890
°286 425
"302 161

"319 211

"337 714
"357 535
"378 824
“401 538
"425 853
"4.51 856
"480 O41
‘510 886

"545 003
"582 861

624 885
"670 399
"720 105
"773 400
"831 208

"894 149
"962 574

3°037 O31

"119 758
‘209 O12

"395 395
"407 823
"515 700
"630 784
°7 52 027
882 729

2°026 872

"193 820
"387 216
"638 272

"958 607

1°397 940

7

Ve) CO oO “I “I On Cn on
0 OO @M~T Ann PWDH BAO OO ON AM PO KH OC LO GW An f$WHN He OC OO COr~ AH

de Oo DS =

Le)
An

113

H¥(5),
Elementary Values—(continued).

AS ES | | yf PD fe

55. | 0°909 744 | 000 587 | 139 | 2°143 O15 | 015 347 | 984.653 | 55
6 | ‘010 331 | 000697 | 144| °158 362 | 017 729 982271] 6
7 | ‘011 028] 000671 | 150] "176091 | 014241 /985 759} 7
8 | *orr 699 | 000 796 | 155] “190 332 | 016 494 | 983 506) 8
9 | *012 495 | 001019] 161 | °206 826 | 021 061 | 978939] 9
6o | 013 514] 001 021 | 169] ‘227 887 | 017 626 | 982 374} 60
I | "O16 535 | 001 201/176] °245 513 | 019 305 | 980695] 1
2 | °015 736] cot 314] 184] °264 818 | 018 483 | 981 517] 2
3 | °017050] 001 453 | 192] °*283 391 | 017729 | 982271] 3
4 | °018 503 | oor 318 | 200] °*301 030 | 010724 | 989276] 4
65 | "org 821 | oor 468 | 205] °311 754] 010 465 | 989 535| 65
6 | *021 289 | oor 425 | 210] °322 219 | 006 161 | 993 839] 6
7 | °022714| O01 Gor | 213 | *328 380 | 006074 | 993926] 7
8 | '024 315 | 001 688 | 216] °334454 | 004.002 1995998| 8
9 | 026003 | 002 182 | 218] °338456 | 007 897 | 992 103] 9
70 | °028 185 | 002 660 | 222] 346 353 | 009 673 | 990 327 | 70
I | *030845 | 003 272 | 227] °356026/] 011 330| 988670] 1
2 | °034 117 | 003 741 | 233] °367356| 009221 | 990779] 2
3 | °037 858 | 004 166 | 238] °376 577 | 005 440/994 560] 3
4 | ‘O42 024 | 003 490 | 241 | °382017 | 990895 | 0C0g105| 4
75 | °045 514 | 004 192 | 236] °372 912 | 990 700 | 009 300} 75
6 | *049 706 | 003 589 | 231 | °363 612 | 978811 | 021 189| 6
7 | 7053 295 | 004 513 | 220] °342 423 | 979 796 | 020204] 7
8 | °057 808 | 005 133 | 210] °322 219 | 976634 | 023 366| 8
9 | °062941 | 005 484] 199] *298 853 | 970 660 | 029 340] 9
80 |} °068 425 | 006 632] 185} 269 513 | 968 533 | 031 467 | 80
I | °075057 | 0070701173] ‘238046 | 960611 | 039 389] 1
2 | °082 127 | 007 127/158} *198657 | 950 562 | 049 438} 2
3 | 089254 | 007 129 | 141] "149219 | 940686 | 059314] 3
4 | ‘096 383 | 006045 | 123} ‘089905 | 927128|072872| 4
85 | °102 428 | 005 449 | 104] °017 033 | 917 465 | 082 535 | 85
6 | °*107877 | 007 207 | 86! 1°934 498 | 916 760 | 083 240] 6
7 | "115.084 | 006159] 71| “851258 | 904617 | 095 383] 7
8 | °121243|009459} 57] °755 875 | 906 883 | 093117; 8
9 | “130702 | 013 441} 46] °662 758 | 905 444] 094 556] 9
gO | “144 143 | 022 805] 37] *568 202 | go8QgrQ | og1 08: | go
I | *166948 | 026 448] 30] °477 121 | 884607] 115393] 1
2 | 193 3961057660} 23] °361 728 | 893 545 | 106455] 2
3 | °251056]069279|) 18] ‘255273 | 823.908] 176092] 3
4) °320335)/ 118998] 12) °079 181 | 765 917 | 234.083] 4
95 | 439 333 7} °845 098 | 756 962 | 243 038 | 95
6 4] °602 060 6

114
HM(5),

Probabilives of Living over, and of Dying tn, a Year, and the
Average

&o © we CS) | bn
S cemoran Hobe O WO GO” Ah AO PWM CO WO COr~3 Ann PS © WH mm O

> © BD fH

>

in
eH be O WO COM Qn

"999 797
"990 828
"99° 743
"99° 775
"99° 569
‘989 998
"989 653
"989 299
"988 935
988 811
988 684
‘988 683
"988 424
988 156
"987 748

‘987 057
"986 341
"985 598
"984 685
"983 733
"982 884
"981 995
"981214
"980 089
"979 959

Tx
(1—pz)

"004 000
°003 414
"003 022
"002 931
"003 O40

"003 253
"003 774
"004 402
"005 45°
"006 720
"008 327
"009 657
"oro 281
"010 709
"O10 825

"010 506
"010 064
"009 943
"O09 704
009 458
"009 203
"009 172
"009 257
"009 225
"O09 431
"010 002
"O10 347
"010 7OI
"O11 065
“ort 189

"O11 316
"OII 317
"Orl 576
"ort 844
"012 252

"O12 943
"013 659
"O14 402
"OI5 315
"016 267
"017 116
"018 005
“018 786

"OIQQII
"020 O41

ation of Infe, at each Age.

el» [ate

48°463 | 55 | ‘977 813 | °022 187
47°650} 6 |°976494| *023 506
46°817] 7 | °974.925| °025 075
45°958] 8 | 973 423 | °026 577
45°91} 9 |°971 640} *028 360
44°227 | 60 | "969 362! *030638
43°370] 1 | °967084| °032916
42°532] 2 | 964417] °035 583
41°718 1 3 | °96r 500| °038 500
40°944 4 | °958290| ‘o41 710
40°218 | 65 | °955 386| ‘044614
39°55] 6 | 952 164) *047 836
38°93%] 7 | "949043 | 050957
38°331] 8 | "945 551| "054.449
377411 9 | 941 882] °058 118
37°148] 70 | 937164 | °062 836
36°537] I | "931 441] °068 559
35°994] 2 | °924.449] °075 551
35°2501 3 |°916520] *083 480
34°600} 4 | "907 769 | ‘092 231
33°926] 75 | "900 506] *099 494
33°236] 6 | °891 854] +108 146
32°539] 7 | 884 514) 115 486
31°838] 8 | 875371] °124 629
31130] 9 | °865085| °134915
30°422 | 80 | 854232! °145 768
29°724] 1 | °841 284] °158 716
29°029] 2 | 827699] °172 301
28°338] 3 | °814230| °185770
27°650| 4 | °800970| *199 030
26°957 | 85 | °789 899] ‘210 I01%
26260] 6 |°780051] ‘219949
25°555] 7 |°767 213 | '232 787
24°848] 8 |°756410] °243 590
24°140] 9 | °740112| °259 888
23°433 | 9° |°757557| °282 443
22°734] 1 |°680851] °319 149
22:042| 2 | 640625 -359375
21°357| 3 |°500974| "439 026
20°681] 4 |°478262] °521 738
20°01 5 | 95 | °363 637 | °636 363
19°354] 6 | ‘000 000 | 1'000 000
18°700

18°049

¥7°405

H™(5),
THREE PER-CENT.

© Cr OAtn HG WY ww O

Sd

&>
min Yw CO WO CHO™* Ain PO WKH | O

&>
YO cn? QU

»
Oo

oO pow x

a
PhO He GO WO CONT A

116
HM(5),

3 PER-CENT.
Commutation Table.

7 440°94 175 322°

7 195°32
6 961°90
6 738°69
6 523°25
6 314°00
6 110°15
5 909°80
5 7%2°41
5 5158:
5 319°16
5 121°23
4.92406
473149
4 544°48

4 364°36
4 192°73
4029°64
3 873°37
3 724°06

3 581°40
3 445°09
3 314°07
3 187°76
3 066°36

2 948°97
2 834'44
2 723°41
2 615°79
2 51°50
241107
2 314°36
2221°52
2 131°85
2 045°24

1 961°34
1 879°57
1 799°90
I 722°31
1 646°53
057257
I 500°63
1 430°70
I 362°93

1 296°89

168 126°9
161 165°0
154 426°3
147 903°0

I41 589'0
135 478'9
129 569'1
123 856°7
118 340°9
113 021°7
107 900°5
102 976°4

98 244°92
93 7 00°44
89 3 36°08
85 143°35
81113°71
77 240°33
73 51627
69 934°87
66 489°78
63.175°71
59 987°95
56 921°60
53 972°63
5113818
43 414°77
45 79898
43 287°47
40 87641
38 562-05
36 340°53
34 208°69
32 163°45
30 202°10
28 322°54
26 522°64
24 800° 33
23 153°80
21 581°23
20 080'60
18 649°90
17 286'97
15 990°08

3 487 413°4
3 312091°2
3 143 964°3
2 982 799°4
2 828 373°5
2 680 470°!
2 538 881-0
2 403 402°2
2 273 83371

233774
2 088°84
2 065°00
2 044°57
2 025°40
2 006°14
1 986°20
1 963°81
1. 938°56

2 149 976°4 | 1 908"33

2 031 635°6
1918 613°9
1810713°4
1707 737°°
I 609 492°1

1515 791°6
1 426.455°5
1 341 312°2
1 260 198°5
1 1829582

I 109 441°9
1039 507°0
973 017°2
90g 841'5
849 853°6

192 932°0
738 959°3
687 821°2
639 406°4
593 607°4

55° 319°9
599 443°5
470 881'5
434 541°0
400 332°3
368 168°8
337 966°7
309 644°2
283 121°5
258 321°2

235 107°4
213 586°2
193 505°6
174 855°7
157 568°7

1 872°34
1 829°34
1 781°32
1732°17
t 682°98
1 635°22
I 590°71
U 549° 74
I 510'84
1474°35
I 440°15
1 408°15
1 377°47
1 347°69
I 319°14
I 291°06
I 262°42
1233°95
I 205°65
1177755
I 15027
1 123°78
1 098°3 5
1.073°38
I 048°87

I 024°54
999°895
974°969
949°801
Q24°192
898°188
872°055
84 5°823
819°730
793°383

75 864°78
73 747°94
7t 65819
69 593°19
67 548°62

65 523°23
63 517°09
61 530°89
59 567°07
57 628° 52

55 720°19
53 847°85
52 018°51
50 237°18
48 505‘01
46 822°03
45 186°81
43 596°r0
42 046°36
49 535°52
39 061718
37 621'03
36 212°88
34 83 5°40
33 487°72
32 168°58
30 877°52
29 615710
28 38115
27 175°49

25 997°94
24 847°67
23 723°89
22 625°S¢4
21 552°15
20 503'28
19 478°74
18 478°85
17 503°88
16 554°08
15 629°88
14 731-70
13 859°64
13 013°82
12 194'09

&& we we = =
pwvwnsd © rr Ar Ro H HO DO Or An PO HHO OHO DO Ahn HO YW =! O

»

tn
hHOobvHme CO OO Cr AM

6

“I ~ AN
Boo wm Rw OHO OO AN PROD HO WO Cr An CH KH O

\o
nn PO HD = O

1232°75
{170°29
1 109°49
1050°17
992°482
936°248
881-129
827°307
774°630
723°113

672°769
624°033
576°876
53 1°534
487°954
446°209
405°991
367°142
329°519
293°214
258°418
22 5°929

117
H¥M(5),

3 PER-CENT.

14757°34
13 587°05
12. 477°56
II 427°39
1043491
9 498°661
8 617°532
7 790°225
7 0157596
6 2927483

5 619°714
4.995°680
4418°805
3 887°271
3 399°317

2 953° 109
2 547°118
2 179°976
1 850°457
T 557°243
I 298°825
1 072°896

141 578°6

126 821°3
113 234°2

100 756°7
89 329°28

78 894°37
69 395°71
60 778°18
52 987°95
45972°35
39 679°87
34 060°16
29 064°48
24 645°67
20 758°40
17 359°08

14 405°98
11 858°86

9 678°882
7 828°425

Commutation Table—(continued).

767°016
7409°462
7313°754
686°743
659°645

632°318
604°469
570°311
5477739
518°775

489°493
460°352
431°370
402°831
374°732

347°199
319°978
292°954
266°024
239°317

6 271°181| 213°061
4.972°356| 188°099

195°627
167°995
142°775

TIQ'OTS
99°451 5
81°230 1
65°275 8
51601 4

40°127 4
30°773 4
23°395 7
17°359 6
12°748 5
9°160 5
6°381 8
4°218 5
2°623 8
1°429 0
"663 5
"234 3

877°269 2
709°27 42
566°499 5

406° 584 7
347°133 2
265°903 I
200°627 3
149°0259
108°898 5
78°125 3
54819 4
37°459 8
24°7 11 3
15°559°7
g'169 0
495° 5
2°326 8
"897 8

"2343
"000 0

3 899°460
3 022°I91
2 312°917

1 746°417
1 299833
952°699
686°796
486° 169

337°143
228°245
150°120
95°300
57°840

33°129
17°578
8°409
3°459
1°332
"234
wetele)

164°378
142°443
122°r16

103°415
86°444 2
7V1I94
5775310
45°7579
35°786 9
27°601 6
21°0302
15°7629
15°6575
8°4408
5°9288
3°9514
2°479 6
1°3612

"6374
"227 4

11 400°71

10 633°69
9 893°228
9 179°474
8 492°731
7 833°085
7.200'767
6 596°298
6 019°987
5 472°257

4.953°482
4 463°989
4,003°637
3 572°267
3 169°436

2 794°704
2 447° 504
2 127°526
1 834°572
1 568°548

I 329°231
1 116°169
928'070
753°693
621°249

499°133
395°718
309°274
238°155
180°624
134°866
99°079
71°477
50°447
34°684
23°027
14°586
8°657
4°706
2°226
865
"227

bw bP H

n

~T
OO GO AM SO He CO LO Or An

~Y

Cc
O

CO
OO woOr~NT Aw HPO HD =

2)
oO

& Db

\O
nn +>

118

H™M(),
3 PERCENT.
arithms and Co-loga arith 0 kiko Nz, and M,; with
3

A
a | LogD, Colog D; (Colog wp.) Log Nz ry
10 |3°871 628)1" "128 372|0°014 578 |5°243 837] 981 Boo
1 | *857 050) * "142. 950] ‘O14 322] *225 637| 981 634
a | 842 728) *1§7 272| *014 152] '207 271/981 450
3 | 828 576) * "171 424) 01g 112) +188 721/981 256
4 | 814 464) * +185 530] °014 160] *169977| 981 053
15 | ‘800 304] “ +199 696] ‘014 252] "151 030] 980 842
6 | *786 052) * +213. 948] °014 479| 131 872] 980630
71 °771 573) ' +228 427| ‘014 753| ‘112 502/980 418
8 | °756 820] * +243 180] ‘015 211| "092 920] 980 215
9 | °741 609} “984 -2§8 391] “O15 766] ‘073 135) 980 027
20 | °725 843] * +274.157| °016468| 053 162/979 861
1 | 709 375| *982 948) -290625) ‘017 052] *033 023/879 715
2 | *692 323] * +307 677] "017 325] ‘012 738/979 572
3 | 674998) * "325 002] °O17 513 14992 310/979 432
4 | °657 485} * "342 515] “017 564] 971 742/979 285
25 | 639921] « +360079| "017 424| °951.027/979 124
6 | 622 497) * "377 503| "017 230] *930 151/978 943
7 | °605 267] “ *394733| ‘017 178] ‘909 094) 978 750
8 | +588 089) * “411 911] ‘017 072] *887 844/978 540
9 | 571017] * +428 983] °016.964| *866 3841 978 310
30 | 554053] * 7] °445.947| °016 853] 844 694) 978 061
1 | *537 200) * +462 Boo} 016 839 *822 7551977 795
2 | °520 363] * +479 639] °016876] “800 5501977 514
3 | 503 485] "496 515| °016862| *778 064) 977 213
4 | 486 623) * *513 377| °016.952) °755 277/976 897
35, | 469 671) * +530 329] ‘017 203 | *732 1741976 571
© | +452 468) “ "547 532] °017 355 “708 745 976233
7 | 435 113] "504 887] "017 509] *684 978) 975 878
8 | 417 oa] 98: 382 396 +017 670| *660 856] 975 506
9 | 399934) * *600 066] 017 724| °636 362) 975 111
40 | °382 210) * 617 790| °017 780} *611 473/974 687
1 | "364 430] * *635 570 °017 780} *586 1601974231
2 | °346 650) “ "653 350| °017 894] *560 3911973 745
3 | °328 756) * +671 244) 018 012] *534 136/973 227
4 | 310744) « "689 256] 018 191] “507 363/972 674
45 | °292 553] "797 447| °018 495] *480037/972 095
6 | 274.058) * “425 942| ‘018 810| *452 132/971 485
7 | 255 248) * “744 752) "019 138] *423 617/970 841
8 | 236 110} * +703 890) "019 540] “394 458970 164
9 | °216 570) * +783 430| "019 960} +364 622] 969 454
§0 | “196610 * +803 390] °020 355] 334.076] 968 7or
x | *176275] * 823 725] 020728 | “302 777/967 900
2 | "1595 547] ° "844 453] °021 073 | °270677| 967 042
3 | "1344741 "865 526] ‘021 572| °237 719] 966 132
4 | ‘112 902] * +887 098] °022 028] 203 851] 965 157

|»

Been S Card aewnd

»

ra
PO PHO © DTAK

een d wave

>

w

ROE HO OOVAN

119

H(5),
3 PER-CENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
t

wr Differences.
z | CologN,

10 |6°756 163

Log M; A Colog M, A

018 200 /3°325 873 | 994 033 14°674 127 |005 967| 10

a]

wo

© <) e
> & Bb» Oo © rT Atn pwe ns OO ON AM HON HO OO CONTA” BOHM EO |W CON AM HPO D &

tn

"774 363
"792 729
"811 279
"830023
"848 970
"868 128
"887 498
"907 080
"926 865

"946 838
"966 977
"987 262
5°007 690
"028 258

"048 973
"069 849
"090 906
"112156
"133 616

"155 306
"177 245
"199 450
"221936
"244 723
267 826
"291 255
"315022
‘339 144
°363 638

"388 527
"413 840
"439 Gog
465 864
"492 637

"519 963
"547 868
"576 383
"605 542
"635 378

"665 924
"697 223
°729 323
"762 281
"796 149

or8 366

025 313
025 769
026255

026 773
027 326

027 905
028 515
029 159
029 836
030 546
031 299
032 100
032958
033 868
034 843

"319 906
"314.920
"310 602
"306 510

"302 362
"298 023
"293 100
‘287 478
"280 654

"272 385
"262 204
°250 743
"238 592
"226 079

"213 577
"201 590
"190 259
"179 218
"168 599
"158 408
"148 649
"139 083
"129 589

"120 290

"TIO 947
"101 205

"og 298
"O81 223
7070 981

"060 799
"050 681
"040 742
"0390755
"020 722
"O10 530
2°999 955
"988 991
977 933
"965 762

"953 367
"940 544
‘927 280
"913 671
"899 483

995 O14
995 682
995 908
995 852

995 661
995 977
994 378
993 176
991 73!

989 909
988 449
987 849
987 487
987 498
988 013
988 669
988 959
989 381
989 809

990 241
99° 434
ggo 506
990 701
999 657

990 258
990 093
989 925
989 758
989 818
989 882
ggo 061
990 013
989 967
989 808

989 425

“680 094
"685 080
689 398
"693 490

"697 638
"791977
"706 goo
"712 522
‘719 346

"727 615
°737 706
"749 257
“761 408
773 921

"786 423
798 410
"Bog 741
820 782
"831 401

"841 592
"B51 351
"860917
"B70 411
"879 710

"889 053
898 795
"908 702
918777
"929 O19

"939 201
"949 319
959 258
"969 245
"979 278

"989 470

989 036 |3°000 045

988 642
988 129
987 605
987 177
986 736
986 391
985 812
985 321

“OLLI 009
"022 367
"034238

"046 633
"059 456
"072 720
"086 329
"100 517

004 986
004 318
004 092
004 148

004 339
004 923
005 622
006 824
008 269

O10 O91
O1I 551
O12 157
O32 513
O12 502

orr 987
OLI 331
OII Og!
o10 619
O10 191

009 759
009 566
009 494
009 299
009 343

009 742
009 907
010075
O10 242
o10 182

o10 118
009 939
009 987
010 033
O10 192

O10 575
O10 964
011 358
o1rr 871
012 395

012 823
O13 264
013 609
014 188
014.679

I
2

3
4

=

~

WO GO~ Qn BO Ww CO WO CON At BPW HN HO }CO Cn QAWr

oO

ir pf & DY

Wee Oo }) Cr~ A

4

120

H™(5),
3 PER-CENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.
A A
z | LogD, (Logep;) Colog D; (Colog op;) Log N; A x

55 |3°090 874'1°977 418 4-Q09 126] 0°022 582 |4*169 008] 964.117] 55
6 | 068 292/ °976 833) °931 708] °023 167] °133 125/963 005
7 | 045 125) °976 134) 954875

8 | 021 259] 975 464! “978 741] °024 536] °057 947] 960 542

9 |2°996 723

60

OS poe ns

~y
hob m CO W/O CONTA

“I
“NAN

oO
©

(oe)
©O G7 At BRO Ym O

2

3
4

"971 391
"945 040
"917 667
"B89 094
"859 206
"827 866
"795 208
"761 082
725 531
"688 379
"649 538
*608 516
564 834
"517 880
"467 185
"412 323
"353972
"291 429
225 296
"154657
078 873

1°997 O11

"9°90 717
‘B14 752
"712 662

"603 441
"488 176
"367 461

"239 540
"105 460

0°961 921

"804 94)
"625 155
"418 922
°155 029

95 {1821 856

6

"369 686

974 668 5°003 277

"973 649
"972 627
"971 42;
"970 112
"968 660

"967 342
"965 874
"964 449
"962 848
"961 159

"958 978
"956 318
"953 046
"949 395
"945 138

"941 649
"937 457
"933 867
"929 355
"924.222

918 738
"912 [06|2°002 389

028 609
"054 960
"082 333
"110906
"140 794

"172 134
"204 792
"238918
"274 469
"211621

"350 462
"391 484
"435 166
"482 120
532 815
"587 677
"646 028
"708 571
774 794
"845 349
"921 127

"905 035] “O90 283
897 10] *185 248
"890 779} °287 338
"884 735) “396 559
879 285) 511 824
°872 079] °632 539
"865 920] *760 460
"856 461) “894 540
"843 020/1°038 079
820214] "195 059
"793 797| °374 845
°736 107] "581078
666 827! *84497)

“547 8300°178 144
| "630 314

025 332
"026 351
"027 373
"028 573
"029 888
"O31 340
"032 658
"034 126
"035 55!
037 152
038 841
"O41 022
"043 682

046 954

"050 695
"054 862
"058 351
"062 543
"066 133
"070 645
075 778
“081 262
"087 894
"094 965
"102 0go
"109 221

"195 265
"120715
"127921
"13.4080
"143 539
"156.980
"£79 786
"206 233
"263 893
"333 173 |1'953 169) 416 517

"452170

3

2

018 489] 959 173

977 662) 957 721
"935 383/956 167
"BOT 5501954 515
"846 0651952 757
"798 822] 950 892

"749 714) 948 881
"698 595/946 710
"645 395/944 340
"589 6451 941 747
"531 392) 938 888

‘470 280) 935 769
"$06 049} 932 403
"338 452) 928 827
'267 279] 925 077
"192 350/921 195

"113 551/917 007
"030 558 912 575
"943 133/907 681
"850 814) 902 386
7153 200| 896 704

"649 904} 890 592
"540 496) 884 227
"424 723/877 667
"302 390| 870 872
"173 262! 863 760

"037 022) 855 769
"892 791) 846 144
738 935| 834 631
"573 566) 819 329
"392 895) 798 856

"IQT 751779 S71

0°962 322] 732 328

694 6501672 103
366 7.53] 586 416

*369 4

6
"023 866| °096 130/961 817| 7
8
9

OS pown

(oo) Co ~
O DIAN PROKDHO OBDVWAN FOKHHO OOO

ve)
co)

Vo)
Aw -& OO WO mo

121

H¥(5),
3 PER-CENT.

their Differences—(continued).

x | Colog N, A

55. |5°83° 992 | |
6 | 866 875 036.995
7 | "903 870/038 183
8 | "942 053 | 039 458
9 | °981 511 | 040 827

Go |4°022 338 042 279
I | ‘064617 1043 833
2 | 108 450/045 485
3 | “153935 | 047 243
4 | 201 178} 049 108

65 | °250286/051 119

~r

~~

co © 2)
Oo o~r QA SFO YP OO OO Om! Am SPW YH ECO YO CGO~™ AN

\O
Oo

Ve)
An PO WD

"BOF 405 | 953 290
"354995 | 055 660 |
"410 355 058 253
"468 608 O61 112

"529 720 | 064 231
"593 951 | 007 597
661 548 071 173 |
"732 721 | 074 923
"807 644 | 078 805

"886 449 , 082 993

"969 442 | 087 425
3°056 867 , 092 319

"149 186
°246 800
"350 096
"459 504
575.277
"697 610
826 738
962 978
2°107 209
"261 065

"426 434
°607 105

097 O14
103 296

109 408

122 333
129 128

136240

144231
153 856
165 369:
180 671

201 144

"B08 249
1037 678
"395 350
‘633 247
0°046 831

630 314

229 429
267 672
327 897
413 584
583 483

oO
"772 969 | 823 784
"596 753 | 797 621
"394 374| 739 554
"133 928 | 670 467

A

869 503 ' 984.04 5
"853 548 983 246
836 794 | 982 517
‘819 311. 981 625
‘800 936' 989 438
"783 374: 979 283
"760 657 | 977 910
"738 567 | 976 412
‘714.979 974 767
"689 746/973 344
°663:090'971 760
"634 850970 273

"605123 968 598

Colog M,

Logarithms and Co-logarithms of D,, N,, and M,; with

"130 497 | 015 955

"146 452

°163 206

016 754
017 483

"180 689 018 375

“199 064/019 562
"218 626 | 020 717

"573 721
"54° 579
"505 120
"466 800

"424.921

36 736

966 8x8

964 541
961 680
958 121

954953

‘378.974 | 949 531
"328 505
°274 387
"215 842:
"153 643
7086 773
"O14 583
115773 \1'9
851 988

"759 902
"660 466

"553 724
"440 934

945 882
941455
937 801
933 130

1927 810

922 153
O15 252
997 914
goo 564
893 258
887 210
881 goQg

"322 843 | 874 704

"066 6o4

"197 637 | 868 967

859 779

926 383 | 846 586

"239 343
"261 433
285 021

"310254
"336910
"365 150
"394 877 |
"426 279

"459 421
"494 880
533 200
575 979
"621 026

"671 495
"725 613
"784 158
"846 357
"913 227
"985 417
2°063 264
"148 012
"240 098
"339 534

"$46 276
"559 066
677 157
"802 363
"933 396
1073 617
"227031
"403 247
"605 626
"866 072

T 804 $95 552 454 0°19 5 605

"356

643 151

' 022 090

023 588
025 233

026 656

028 240
029 727
031 402
033 142

035 459 |.

038 320
o41 879
045 947

050 469 | -

054 118
058 545
062 199
066 870
072 190

077 847
084 748
092 086

099 436
106 742

112 790
118 og!
125 206
131033
140221
153 414
176210
202 379
260 446
329 533

447 546 :

& © bY -

~y ~Y AN
Oo Or Am BHO Hew! QO HO CON AWN PhWO DW =| CO LOO CO~™ QAN

CO

CO

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‘Oo
An

bone La

J

i>)
Oo conT Qh pu pHs © GO An FPO HK CO OO GO Qh HO YO HO

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aN
Read

in
WH me CO WY} Cnr Qn

23°366 2
23°1496
22°916 3
22°673 2

22°424 7
22°1728
21°924 5
21°682 1
21°4549
21'248 1
21°069 2
20°9129
20°764 I
20°618 5

20°469 4
20°307 4
20°129 2
19°943 4
19°740 9

19°527 3
19°2999
19°062 9
18-818 2
18°563 3
18°302 2
18°041 7
17°777 3
17° 508 7
17°2356
16953 6
16°662 I
16°3.58 4
16°046 5
15°7260

15°398 7
15°068 6
14°735 6
14°399 5
Ig°062 2

13°723 5
13°381 4
13°035 6
12°683 7
12°329 6

122

H™(5),
83 PER-CENT.

Assurance of a Unit.

ad ay A;

23°561 9|°284 607 |
"290 306 |°
"296 614 |°
"393 497 | °
"310 489 |:
"317 730)"
"325 066 |°
332 298 |:
339 358 |:
"345975 ]°

"3.51999 |°
357 206°
"301 7590|°
"366095 |°
"379 334

"374 676
"379 396
"384 585
"390 058
"395 896

"402 119
"408 742
‘415 645
"$22 770
"$30 197
"437 800
"445 387
"4.53 091
"460 913
*468 864
477 078
"485 569
"494 416
"593 500
512 836

"522 369
"531 983
"541 680
551471
"561 296

"571 159
“581 124
"591 199
"O01 4.46
‘O11 760] °

"O17 130

"O17 4.52
"017 806
"018 202
018 626
"019 088

"O19 590
"020 135
"020 717
"O21 332
"021 ggo

"022 681
"023 390
"024 130
"024 903
"O25 711
"026 573
"027 492
028 483
"029 537
"030 661

"031 854
"033 107
"O34 424
"035 811
"037 265
‘038 792
‘040 408
O42 121
043 954

O45 895

—_——__

oa
© cons Qn

NS pwnd

“I
WO Cn~r Qin SPW BHO WHO CONT AWN

CO oo “I
BS wo aorak & © YB! O

pf & DH mm

\O
Onin

ay

Il‘Q71 1

I1'6100
11°246 2
10°881 5
IO°5140

TO'I45 4
9°780 1
9°416 4
97056 7
8-701 9
8-353 1
8005 5
7°6599
7°313 3
6°966 5
6°618 2
6°273 8
5°9377
56156
573109
5°026 1
4°7488
4°484 4
4'2220
3°967 8

3°7 242
3°490 §
3°273 5
3°973 5
2°888 o

2°713 8
2°5387
2°3522
2°1579
17938 4
1°697 6
1°4368
1173 5

"886 8

628 3

"353 1
"000 0

"622 200
632 719
"643 314
653 936
"664 643

075 375
"686 016

696 610
‘797 087
"717 420

‘727 579
"737 794
‘747770
"757 865
"797 966
"778 ILO
"788 142
"197 932
"807 312
"816 186

"824 484
832 559
"B40 259
‘847 904
B55 307
"862 402
"869 210
"B75 530
"881 354
"886 756
‘891 832
"896928
"902 364
"908 024
"914 416
"921 430
"929 026
"936 694
"945 O44
"952 575

"960 592
"970 874

Values of Annuities, and Single and Annual Premiums for

Wx

"047 968
"050 176
052 532
7055 038
"057 725
“060 596
"063 637
"066 877
"070 310
"073 946
"077 790
"081 917
"086 349
"O91 163
"096 400
"102 138
"108 353
"IIS O14
"122031
"129 329
"136 820
"144 823
"¥53 209
"162 372
"172171
"182 551
193 567
"204 877
"216 361
"228 o74

"240 139
"253 402
"269 186
"287 543
"311 199

"341 577
"381 257
"43° 954
"500 869
585 026

‘799 947
"970 874

HM(5).

THREE AND A HALF PER-CENT.

124

HM(5),
33 PER-CENT.
Commutation Table.

N- S; M,;

bot
bo

b=
OO G3 An ff © YH = OC

7089°1g |152122°0 |2851 473°6| 1 705'23

6 822°06 | 145 300°0 |2 699 351°6| 1 677°84
6 568°86 |138 731°! 12 554051°6| 1 655°34
6 327°54 |132403°6 |2 415 320°5| 163615
6095°65 |126307°9 |2282917'0| 1 618'24

57 400°61
55 95°38
54 017° 54
52 362°21
50 726°06

49 107°82
47 5°7°49
45925°61
44 364°35
42 82624

41 31570
39 837°82
38 398°79
37 002°93
35 651°03

34 342°92
33 977° 13
31 850°59
30 660°00
29 503°37
28 378°46
27 283°11
26 215°31
25 173°78
24 157°63
23 165°71
22 197°48
21253°31
20 332°93
19 436°10

ome

§ 87161 11204363 |2 156 6og'1| 1 600°33
565460 | 1147817 | 20361728! 1 581°88
5 442°77 | 109 3389 | 1921 3911 | F 561°26
5 235°56 | £04 103°4 | 58120521) F 53811
5930°95 | 99 972°42) 1 707 948°8| 1 510°54
482815 | 94244°27| 1 608 876'4| 1 477°88
4.626°03 | 89 618°24) 1 514 632°1| 1 439°03
442644 | 85 19180) 1 425 013°9/ 1 395°87
4232°78 | 80959°02| 1 339 822°1|/ 1 351°90
4045°85 | 76913°17|1 258 863°0| 1 30810

3866°72 | 73 046°45|1 181 949°9| 1 265779
3696°71 | 69 349°74| 1 108 903°4| 1 226°54
3 535°79 | 05 813°99| 1 039 553°7| 1 190°59
3 382°22 | 62431°77| 973 739°7| 1 156°63
3230°13 | 5919563) O11 307°9|1 124°9)
3097°13 | 5609851) 852 112°3| 1:095°34
296486 | 53 133°65| 796013°8| 1 067°81
2838°32 | 50295°33) 742 88071| 1 o41'53
2716°95 | 47578°38| 692 584°8| 1 016°14
2600°86 | 44977°52| 645006'4| 991°927
2489°21 | 4248831} 600028'9| 968°229
2 380°98 | 40107733) 557540°6| 944°173
2276°66 | 37830°68| 517 433°3] 920°370
217613 | 35654°55| 479602°6| 896°831
2079°28 | 33575°27| 4439480) 873°566

iS)
we

&>
WO x23 Ain FH HMO OO CO~™ Qn + © BW HO

YS

Go

Go
WO DWr~3 Ann POH HO LO OAH HPO HHO OO OAM HPO WH O

Oo

40 |1986°48 | 31 588°79| 410372°8| 851°087| 18 562°54 | 40
1 |1897°59 | 29691°20| 3787840] 829°368)17711'45 | 1
2 |1812°67 | 27878'°53) 349092°8} 808°619|16882°08 | 2
31173010 | 26147°44| 321214°2| 788°345] 16073°46 3
4|1652°75 | 2449469] 2950668) 768'535|15 28512 | 4

45 11577°29 | 22917°39| 270572°5| 748°971/ 14 516°58 | 45
6 }1504°23 | 21.413°16| 247654°7| 729°246| 13 767°61 6
7 |1433°51 | 1997995) 226241°5) 709°304 7
8 |1365°09 | 18614°57| 206261'9| 689°446 8
Q {129872 | 1731584) 18764773) 669°246 9

go | 1-234°39 | 16081°45|] 170331°5| 648°834 50
I }1172°24 | £4909'21; 154250°0] 628°420 I
2 {1112°20 | 13797°01| 139340°8| 608°028 2
3}1054'41 | 1274260) 125 543°8] 587°841 3
4 | 998°466| 11744713) 1128012] 567°557 4

in
© GON Ati

pwnd

wl ~A

“I

5
6
7
8
9
fo)
I
2
3
4
5
6
7
8
9

foe)
OO @O~ Qn APO HD ww

Ve)
Oo

a® PO DH =

34°3557
26°587 4
20°2Q1 2
15°2929
11°336 1

8°284 8

5924 3

4°107 3)"

2°7O1Qg
1°672 4
"906 4
"4189
"147 2

125
Hs),

34 PER-CENT.
Commutation Table—(continued).

10 799°634
9 9077321
9 0657448
8 272°441
7 526°614
6 826°444
6 170°678
5 557°943
4 986°994
4456°590
3 955°499
3 512°182
3 095°147
2 712°746
2 363°395

2 O45°474
1 757°005
1 498°540
1 267°146
I 062°240

882°523 8
726°160 5
591°423 2
476°276 2
378°888 4

297°488 7
230°3059
175°697 4
132'0264
97°0707
71°083 3
50°792 2
35°499 3
24°163 1
15°878 3

9°9540
5°846 7
3°144 8
1°472 4

"5660

"147 2
"000 0

Sz

101 057°09

9° 257°45
80 350°13
71 284°68
63 O12°24

55 485°63
48 659°19
42 48851
36 930°57
31943°57
27 486°98
23 521°48
20 009'30
IO QI4°15
14 201°41
11 838-01
9 792°539
8 034°934
6 536°395
5 269°249
4 207°008
3 324°485
2 598°324
2 006901
I 530°625

1151°736
854°248
623°942
448°244
316°218
218°547
147°404
96°672
61°173
37909
21°13!
11°l77
5°330
2°186
"713

"147
°600

M,

547°355
527°108
506'843
486°446
466°083

445°646
424°920
404°065
382°999
361°761

340°386
319°217
298°266
277°734
257°O16

237°999
218°698
199°629
180°719
162°055

143°796
126°519
110°181
95°147 2
81°281 8
68-5870
5771228
46°820 4
37°729 5
29°891 I

23°284 5
17°887 4
13°575 3
10°1357

7°4077

5°387 4
3°7797
2°504 2
1°5660

"856 6

"399 7

"1422

R;

7 929°600

7 382°245
6 855°137
6 348°294
5 861°848

5 395°765
4950118
4 525°198
4 121°133
3 738°134

3 376°373
3 035°987
2 716°770
2 418° 504
2 140°770
1 883°154
1 645°154
I 426°456
1 226°827
1 046°108

884°053 3
740°257 8
613°738 3
593°557 1
408°409 9
327°128 1
258°541 1
201°418 3
154°5979
116°868 4

86°977 3
63°692 8
45°805 5
32°2302
22°094 5
14°626 8
9°239 4
5°468 7
2°964 6
1°398 5
"5419

"1422

Ve) CO foe) ~ ww A nN in
0 OO ONT AN BO DMO WO GO QA PW HD Hw CO OO DO~X QAM BO HD | O WO Cr Qn

ak f& WY

126

HX(5),
31 PERCENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

Colog D

wo

fe . Go Qo
OW YW tO OO COA? Ain PWN | O OO COr~3 Ar DW WO mum

w=
POH HO HO Or An

*626 626! -
"607 O10} °

"587 343
"567 815
"548 482
"529 202
"510 027

"499 959
"472 004
"453 061
"434 082
“$15 117
*396 061
"379755
"357 298
"337 685
"337912
298 085
"278 202
"2.58 319
238 321
*218 207

"197 913
"177 314
"156 401
"135 160
"113 517
"OOT 454
"069 016
046 185
"023 008
2°999 333

a OS

"980 472
‘980 667
"980 720
980 825
"980 932

"981 045
"981 057
‘981 021
"981 035
"980 9.44
"980 604
"980 543
"980 387
980 227
"980 173

"980 117
"980 117
"980 002
"979 886
"979 706

"979 401
"979 087
978 759
978 357
"977 937

"977 562
"977 169
"976 823
976 325)

"975 869]3°000 667

"231 242
"247 598
"264 180
"281 037
"298 350
"316 219
“334 791
"353.946
"373 374
"392 999

"$12 657
"432 185
"$51 518
“$70 798
"489 973

"509 O41
"527 996
"545 939
"565 918
"584 883

"603 939
623 245
"642 702
‘662 315
682 088

"701 O15
"721 708
"741 O81
*761 679
‘781 793

802 087
"822 686
"843 509
"864 840
886 483

"908 546
"930 984
"953 815
976 992

A
* (Colog vp.) Log

"016 426
016.255
"016 215
"016 262

"016 356
016 582
"016 857
"O17 313

“018 572
‘O19 155
“O19 428
‘org 616
‘o1g 667

‘019 528
"O69 333
"o19 280
"O19 175
‘o19 068

"018 955
018 943
"018 979
018 965
"019 056
"org 306
"019 457
"019 613
"019 773
"o19 827
org 883
"org 883
“o19 998
"020 14
020 294

"020 599
"020 913
"O21 241
"021 643
"022 063

"022 438
"022 831
023 177
023 675
"024 13]

3°850 597|1°983 31914°149 403/0°016 681 |5°182 192) 980074| 10

; , . "162 266] 979 908
"142 1741979 726
"121 900/979 531
‘101 431/979 326| 4

"080 757/979 116
"059 873] 978 go2
"038 7751978 690
"017 465
"017 869 /4°995 953] 978 302

"9742551978 141
"952 396] 978 002
"930 398] 977 867
‘908 265/977 736
°886 0011977 598

"863 5991977 446
"841 045/977 273
818 318) 977 088
795 406| 976 884
"772 290] 976 661

"748 951/976 419
"725 3701976 158
"701 5281975 882
677 4101975 586
"652 996] 975 274
"628 270) 974 954
"603 2241974 620
"577 844/974 271
"552 1151973 995
"526 020) 973 513

"499 533|973 095
"4.72 628] 972 642
"445 270) 972 159
"417 429) 971 643
"389 072) 97 1 093
"360 165/970 516
330 681) 969 907
300 588
"269 853| 968 591
238 444] 967 881
206 325| 967 130
"173 45 5| 906 330
"139 785} 965 473
"105 2581964 563
"069 821| 963 588

978 488

969 265

|
O COr~1 Qn

2

©)

ey
© oN Qn PS © DN mn

fe)

3

3

O Cr Attn PO DW

Oo

4

tr of OO DW om

4

5

GO be oO WO COn A

4

127

HX(5).
: 31 PERCENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

I | RS | Se | Ra

992 966 |4°768 216 | 007 034

Ln |

O19 926 |3°231 784

rey bet
OO COo~r QA —& WH HD m

S

> ~ &o o
BoP ew O LO CONT Ath BH Bm O CO DWT QAN SW BW KH O

CA
BH Hm OO / Dr Qn

"069 602
"091 735
°113 999
"136 401
"158955
"181 682

"204 594
"227 710

"251 049
"274 630
298 472
"322 590
"347 004

"371730
"396 776
"422 156
"447 885
‘473 980

"500 467
“527 372
554730
*582 571
610 928

"639 835
"669 319
"699 412
"730 147
‘761 556

793 975
"826 545
"860 215
"894 742
"939 179

020 092
}]020274
020 469
020674

020 884
021 098
O21 310
O21 512
021 698

021 859
021 998
022 133
022 264
022 402

022 554
022 727
022 O12
023 116

023 339

023 581
023 842
024 118
024 414
024 726

025 046
025 380
025 729
026 095
026 487

026 905
027 358
027 841
028 357
028 go7

029 484
030 093
030 735
031 409
032 119

032 870
033 670
034 527
035 437
036 412

2

224 750
"218 886
"213 824
"209 042
"204 210
"199 173
"193 475
-186 988
179 133
"169 638
"158 070
"144 844
"130 O44
"116 642
"102 361
088 682
"075 704
063 193
“OSI 119
"039 55°
"028 492
"017672
"006 956
"996 480

"985 978
"975 95%
"963 963
952 711
"941 296

"929 974
"918 747
"907 744
"896 716
88.5 664

"874 405
"862 874
"850 888
"838 500
"825 586
"812 134
"798 250
783 924
"769 260
"754 999

994 136
994 938
995 218
995 168

994 963
994 302
993 513
992 145
99° 505

988 432
986 774
986 100
985 698
985 719
986 321
987 082
987 429
987 926
988 431
988 942
989 180
989 284
989 524

775 250
“781 114
-786 176
79° 958
795 79°
"800 827
"806 525
"813 012
"820 867

"830 362
"B41 930
"855 156
"869 056
‘883 358

"897 639
"911 318
"924 236
"936 807
948 881

"960 450
"971 508
‘982 328
"993 O44

989 498 |3°003 520

989 074
988 o11
988 748
y88 585
988 678

988 773
988 997
988 972
988 948
988 8or

988 409
988 014
987 612
987 086
986 548
986 116
985 674
985 336

984 749
984 260

"OF 022
"024 948
"036 037
"047 289
058 704
"070 026
"081 253
"092 256
"103 284
"IT4 336
"125 535
"137 126
"149 112
"161 500
"174.414
"187 866
"201 750
"216076
*230 740

"245 991

005 864
005 062
004 782
004 832

005 037
005 698
006 487
007 855
009 495
orr 568
013 226
013 goo
OI4 302
014 281

013 679
012918
O12 571
O12 O74
Orr 569

011 058
o10 820
O10 716
010 476
O10 502

010926
011 089
O11 252
OII 415
OI 322
OIl 227
OTI 003
O11 028
O11 052

O11 199

O11 59!
o11 986

o12 388
O12 O14
O13 452
013 884
O14 326
O14 664
O15 251
O15 740

F eo &o Y) i) md
PwWbH HO HO DON AN SPOON HO OHO DAM PO HM OO DO Orn Qn HO HH OC

pb
OO G~1 Aun

in
Oo

-— O& bw &

128

HM(5),
33 PER-CENT.
Logarithms and Co-logarithms of D,, N,, and M,; with

their Differences—(continued).

&

Log D, Colog D, Log N,;

A
(Log vpz)

On tn
fw Ym O © ow of |

“vn

~J ~?
Mm Ob wm O OO Dr? QA BOND HO OO O~ Qn

1° @)

‘oO CO
PO Dm O WO CON? A

AR

-_
e O) e e °

"974730
"974.030
973 361
"972 565
"971 546
"970 524
"969 324
968 009
"966 557
"965 238
"963 772
"962 345
"9650 745
"959 056

"956 875
"954215
"959 943
“O47 202
"943 035

"939 546
935 353
"931 765
"927 251

"049 483
"074 753
"100 723
"127 362

"154 7971
"183 251)
"212 727;
"243 403.
"275 394
308 837
"343 599
379 827
“417 482
456 737
"497 681
"540 806
"586 591
"63.5 648
*688 446
"745 411
"B05 865
"870 512
938 747

"922 119|2°o11 496

"916 635
"910 003
‘Qo2 932
"895 806] °359 807
"888 677

882 631
"897 182
"869 976
"863 817
"854 358)1°081 718
840916] -227 360
818 112) °386 444
"791 663! °568 332
"734. 004) °776 669
"664. 72510°042 665

"545 726] °377940

‘089 377
"172 742
"262 739
"464 OO1
575 324
692 693
815 51)
"9045 535

"832214

A
(Colog vp,)

"025970
026 639

"027 435

"028 454
"029 476
"030 676
"O31 QQ!
"033 443

034 762
"036 228
"037 655
"039 255
"O40 944

"O43 125
"045 785
"O49 057
"052 798
"056 965
"060 4.54 |2
"064 647
068 235
"072 749
"077 881
‘083 365
"089 997
"097 068
"104 194
"ITI 323

"117 369
"122 818
"130024
"136 183
"145 642

2°975 202/1°97 5 315/3°024 798) 0°024 685 |4°033 4009] 962 5.47
"025 279 13°995 955) 961 433
957 389] 960 245
"917 6341958 966
"876 6001 957 595

"834 1951956 138
790 3331954 581
"744 914] 952 925
"697 839) 954 164
"649 003] 949 295

"598 298) 94.7 279
"545 5771945 104
"490 681] 942 728
"433 409| 940 127
"373 536/937 258

"310 794/934 127
"244 9211930 747
"175 668) 927 159
"102 827] 923 396
026 223/919 504

045 727
"861 033/910 865
"771 898) 905 961
"677 859, 900 652
578 511| 894 960
"473 471| 888 834
"362 305] 882 460
"244 765] 875 896
"120 661] 869 103
“989 764) 862 oc04
851 768] 854 029
"795 797| 844 423
"550 220) 832 933
383 153) 817 652
"200 805] 797 192
"159 084 |0°997 997) 768 914
"181 888) °766 911! 730 683
‘208 337| 497 594,670 444
"265 996 _*168 038) 584 786
335.275 /1'°752 824! 414.962

454274 °167 78

915 306

un
© Or An

in © WL

A

ee) fo 2) aj wr
Oo cm An #+#wON AO OO CO~™N AN HON AO WO COXA A

‘Oo
oO

‘© .
An bwWH Hm

Logarithms and Co-logarithms of D,, N,, and M,; with

az | Colog N
55
6
7
8 | *082 366
9 | °123 400
6o | *165 805
I | "209 667
2 | °255 086
3 | °302 161
4 | 35° 997
65 | ‘401 702
6 | “454423
7 | °5°9 319
8 | °566 591
9 | 626 464
70 | *689 206
I | °755 979
2 | 824 332
3 | 897 173
4 | 973.777
75 |3°°54 273
6 | °138967
7 | *228 102
8 | °322 141
9 | “421 489
80 | °526 529
1 | 637 695
2 | °755 235
3 |_°879 339
4 |2°010 236
85 | °148 232
6 | °294 203
7 | 449 780
8 | 616847
9 | "799 195
go |1°002 003
r | *233 089
2 | °502 406
3 | 831 962
4 |0°247 176
95 | “832214
6

129

HM(5).
31 PERCENT.

their Defferences—(continued).

A

EE

a ,

A

Colog M,

ES |

037 453 |2°738 269 | 983 631 |3°261 731

038 567
039755
O41 034
042 405
043 862
045 419
047 975
048 836
050 795
052 721
054 896
057 272
059 873
062 742
065 873
069 253
072 841
076 6o4
080 496

084 694
089 135
094 039
099 348
105 040
111 166
117 540
124 104
130 897
537 996

145971
155577
167 067
182 348
202 808

231 086
269 317
329 556
415 214

‘oh

"721 goo
"704 873
‘687 035
668 463

648 OQ
628 307
"606 451
"583 198
"558.422

531972
"504 087
"474 004
"443 629
"410974

| 982 973

982 162
981 428
' 980 528

979 316
978 144
| 976747
975 224
973 550

972 115
|97° 517
969 025
907 345
965 602

"376 576 963 269

"339 845
"300 224
"257 003
"209 663
"157 745
"102 157
"042 108
978 396
"909 993
"836 242
"756 810
"670 435
576 681
“47 § 541
"357 067
"252 547
°132 749
"005 853
873 186

731 378

"576 421
"398 665

"194797

585 038 |1'932 798

"601 746
"152 846

960 379

956779
952 660

948 082

944 412
939951
936 288
931597
926249
920 568
913 625
906 246
898 860
891 526

885 480
880 202

873 104| ‘8

867 333
858 192

845 043
822 244
796 132
738 oo1

°278 100
"295 127
"3129065
"331 537

"351 009
"371 693
"393 549
*416 802
"441 578
"468 028
"495 913
"525 396
556 371
“589 026

"623 424
660 155
"699 776
"742 997
"79° 337
842255
"897 843
"957 892

2°02 1 604

I

"O90 007

"163 758
"243 190
"329 565
"423 319
524 459

"632 933
747 453
7251
994 147
"126 814

"268 622
"423 579

601 335
"B05 203

668 948 |0°067 202
551 100] °398 254

"847 154

073751
079 432
086 375
093 754
101 140
108 474

114 520
19 798
126 896
132 667
141 808

154957

177 756
203 868

261 999
331 052

448 goo

O
A pPopne 6 woo

‘oO fo 2) fo.) ~Tr
PWPH OE OBDNAN BWOHHO ODIO FwWKHO OOD cr

\o
Qn

130

HM(5),
33 PER-CENT.
- Values of Annuities, and Single and Annual Premiums for

x a, A; x | A; @,
I
IO |21°458 3 |°240 S40] ‘oro 71tT| 55 | 11°4342.°579 519 "046 607
I |21°298 6|°245 943|‘o11 030] 6 | r1°r03.0'*590 722, '048 808
2 |21°119 5|°251 997 |-011 393] 7 | 10°7682 | ‘602 o413, O51 158
3 |20°9250}°258 579 |o1r 7QO4] 8 | 10°4317 .°613 420 °053 659
4 |20°7210|°265 474)°012 222] 9 | 10°09! 6 *624.922 °056 342
15 |20°511 6/°272 554] "0126701 60 | 9°7497 °636.484 *059 209
6 |20°298 8|°279 751] 013.135] 1 | 9'40991°647 974! 062 246
7 |20°088 9 |°286 850|-or3 602} 2 | g'0707 *659 444 °065 481
8 | 19'883 9|°293 782}'014 067] 3 8-734 6 |°670 812 *068 g10
9 | 19°6926}°300 250/*org 510] 4 | S402 3 | 682 050 °O72 541
20 |19°5197|°306 096|°014.917] 65 | 8°0749'*693 122. 076 378
I 119°372 6)°311 072|"015 269] 6 | 7°7478 "704 184 "080 499
2 |19°2462/°315 348/015 576] 7 | 7°4218 °715 205) 084 923
3 |19°126 7 |°319 388|"015 869] 8 | 70940 |°726 292 ‘089 732
4 | 190104 | "323 319)"016 158] 9 | 6°7651,°737 414 "094 965
25 |18°891 0|°327 354] °016.457] 70 | 6°433.91°748 612 *100 702
6 |18°75991°331 793|"016 791] 1 | 61056) °759715 "106918
7 |18°613 8|°336 730/-017 168] 2 | 5°7844' "770 575] "113 580
8 | 18°458 81 °3413 972|°017 5741 3 | 5°4762) °781 000, °120 597
Q | 18'292.1]°347 Gro|o1B8018F 4 | 5°1841 *790 877,°127 890
30 | 18°153.11°353 664] °018 5041 75 | 4°Qt06 “800 aa "135 370
I | 17°9212|°360 154|*019 034} 6 | 4°6441 °809 137, “143 361
2 [17°7201|°366 953|"019 Go2z] 7 | 4°3895 °817 749; "151 731
3 | 17°511 7 |°374 002|"020 204] 8 | 4°1362 °826 310, "160 879
4 |17°293 3|°381 384|°020 848] 9 | 3°8905 (834 S20 "170 661
35 |17°069 0|°388 971 | *021 527] 80 | 3.6547 “842 596 “18k 022
6 |16°844.9|°396 549|°0222227 1 | 3'4280'°850 260) "192 017
7 |16°616 7 |*404 264|°022 948] 2 | 3°217 4/°857 382! .203 297
8 | 16°384 4|°412 122]°023 706] 3 | 3°023 2|°863 948 °214 742
Q |16°1476|°420 130] ‘024 501] 4 | 2°8429 °870046!°226 402
40 |15°90T 9|°428 439) "025 349] 85 | 2°673 6,875 772/°238 399
T | 15°646 8 | °437 064)°026255] 6 | 2°503 2 "881 536) "251 639
2 115°379 8] °446 093 |°027 234] 7 | 2°3213 °887 688) °267 271
3 |15°1046/°455 402)°028 278] 8 | 2°131 5!°804 104 285 518
4 |14°8206|°465 004 |°029 392] 9 | 1°9166 *Q01 372/°309 053
45 |14°529 5|°474 846 |°030 577] go | 1°680 2) “gog 364) "339 292
6 | 14°235 3|°484 797 |°031 821] 1 | 17423 5 °918048)*378 813
? 113°937 6|°494 866|°033 129] 2 | 1°163 9'°926 824) "428 306
8 | 13°63621°505 057|°034. 507] 3 "B80 5 | "936 410) °497 970
9 | 33°333 9)°515 310/°0359531 4] °6244/°945 068 °581 782
50 |13°027 8|*525 630) "0374711 95 | 3513 +954 303] *706 190
1 | 12°7186/°536 085 |°039 077} 6 | “0000 "966 184) "966 184
2 |12°405 1|°546 688 | ‘o40 782
3 | 12°085 11 °557 509 | "042 606
4 |11°762 2|°568 429 | "044 540

Assurance of a Unit.

HX(5),
FOUR PER-CENT.

OO we iS) ~_
popud Oo oar BO Wm O OO CO~ Air BOON HO OO CON At HPO DW | O

as

in
- i Hb mw O OO Cn An

6 755°64
6 469°83
6 199°75
5 943°28
5 997°95

5 462°14
5 234°97
5 014°63
4800°53
4 59°°74
4 384°51
4 180°77
3981715
3 788°67
3 603°O4

3427°82
3 26135
3 104°36
2.955°28
2 814°04
2 680°21
2 553°41
2 432°68
2 317°47
2 207'78
2 102°84
2 OOI'74
I 904°83
1 811°97
I 723°00

1 638°19
1 557°30
1 480°52
T 407°0g
I 336°95

1 269°78
I 20514
I 142°96
1 083°17
1 025°56

970°073
916°797
865°664
816°732
769°683

132
HM(5),

4 PER-CENT.
Commutation Table.

132 786°0

126 316°1
120 116°4
114.173°1
108 47 5°2
103 013°0
97 778°06
92 703°43
87 962-90
83 372°16
78 987°65
74 806°88

70 825°73
67 037°06

2 348 401
2215615
2 089 298
1 969 182
1855 009

1746 534
1 643 521
T 545 743
1452979
I 365 016
1281 644
I 202 657
1127 850
1057 024

63 433°11
60 005°29
56 743°94
53 939°59
50 684°31
47 870'27
45 190°06
42 636°65
4.0 203°96
37 886'50
35 678°72

33 575°88
31 574/14
29 669°31
27 857°34
26 134°34
24 490°14
22 938°78
21 458°27
20051°I7
18 714°22

17 444°44
16 239°31
15 096°35
14.013'18
12 987°62

12 017°55

11 100°75

Io 235°09
9 418°353
8 648°671

989 986'9
926 553°8
866 548°5
809 804" 5
756 164°9
705 480°6
657 6104
612 420°3
569 783°7
529 $79°7
491 693°2
456 014°5
422 438°6
390 864°4
361 195°!
333 3378
397 203°5
282 707°3
259 768'5
238 310°3
218 259'1

199 544°9

_ 182 1004

165 861°1
150 764°8
136751°6

123 764'0

111 746°4

100 64.5°7
90 410°6!
80 99226

1 388°66
I 362°67
I 34144
I 323°42
¥ 306°67
I 290°02
I 272°93
1 253°94
1232°71
1207°55
1 177°89
1 142°78
I 103°96
1 064°61
1025°59
988-082
953°456
921°896
892°216
864'640

839°050
815°333
792°813
771160
750°603

730° 583
710°359
690°443
670°844
651°564
633°027
615°202
598°255
581°775
505°751
5 50°00]
534° 198
518°370
502° 541
486° 590

470° 549
454° 584
438°712
423°07 5
407°438

43 851°53
42 462°87
41 100°20
39 758°76
38 435°34

37 128°67 -

35 838°65
34 505°72
33 311°79
32 079°08

30 871°53
29 693°64
28 550°86
27 446°89
26 382°29

25 350°69
24 368°61
23 415°16
22 493°26
21 60104

20 736°40
19 897°35
19 082°02
18 289°21
17 518°05
16 767°44
16 036°86
15 326°50
14 636°06
13 965°22
13 313°65
12 680°62
12 065°42
11 467°17
10 885°39

10 319°64
9 769°640
9 235°442
8 717°073
8 214°531

7727941
7 257°392
6 802°809
6 364°097
5 941°022

& i) N ba i
% © err an hWPHHO WO OWTAN BOHN HO 1 BYAN BW DW =O

OO YD

a

e&wWeoeme O WO Cr Qn

&

wn
Oo CON Qh

co,

>,
OO ON An HOOK mM O YO O~ Aan POO PB KH O

~I

~

eo)
Oo

co
YO Cor Cin POOH =

Oo

Ve]

— GO bP

Le)
On

Dz

724° 582

681°256
639°656
599°631
561°244
524°353
488°738
454°472
421°443
389°632
3 59°920
329810
301°955
275°547
2.50°523
226°887
204°452
183°Ir!
162°766
143°441
125°203
108°4.10

92°966 8
79°067 8
6675516

55°358 4
45°4701
36°7820
29°273 5
22°918 6
176511
13400 3
10°055 4
7417 92
5°395 18
3°839 47
2°649 07
1°73425
1068 28
"576 23

"26499
"092 65

133
HX(5),

4 PER-CENT.

Commutation Table—(continued).

Nz

7 924089

7 242°833
6 603°178
6 003°547
5 442°303

.[4917°950

4429211
3 974°739
3 553°296
3 163°663
2 804°644
2 474833
2172°878
1 897°332
t 646°809

I 419°922

1 215°470

1 032°359
869°592 8
726°152 1

600°949 2
492°539 6
399°572 8
320°5050
253°953 5

198°595 1
153°1250
116°342 9
87°069 4
641509

46°499 8
33°O93 4
23°038 0
15°620 J
10°224 9

6°385 5
3°7304
2°002 I
"9339
"3576
"092 7
“9000 O

8;

72 343°59

O4 419°50
57 176°67
5° 573°49
44 569°94
39 127°64
34 209°69
29 780°48
25 805°74
22 252'44
19 088°78
16 284°13
13 809°30
11 636°42
9 739°092
8 092°283
6 672°361
5 450°892
4.424°533
3 554-940
2 828°788
2 227°839
1 735°299
I 335°726
1015221
761°268
562°673
409°548
293°205
206°135

141'°984
95°485
62°391
39°353
23°733
13°508
77123
3°386
1°384
"450
"093
“000

M;

391941
376°483
361°085
345°662
330°339

315°034
299° 587
284°118
268°569
252°967

237°340
221'°939
206°769
197°074
177°548
163°548
149°840
136°362
123°060
109°995

2 307°800
2 070°460
1 848°520
I 641°751
1449°777

1 272°228
T 108°680
958°840
822°478
699°418

589°424
492°150
406°853
332°831
269°131
214°906
169°316
131°484
Too'5Q1
75°792 3
56°222 6
41°038 8
29°4209
20°638 4
14°106 5

Q°3121
58659
3°462 4
1°8719

"880 6

"340 3
"089 I

wr
YO O~r Qn

3

oO po wm

oo ~r “sr
"nr FW RHO CH Cr Ahn PW He CO OO Cr A

c

ak poond O coxnr A

134

H¥(5),
4 PERCENT.
Logarithms and Co-logarithms of D,, N,, and M,; with
their Differences.

Colog D, Log N, A

A A
(Log epz) (Colog vp.)

bmi

~

o tS)
OO CONTA POKP HH O OH ONAN PO HH CO DO GO AN HW HW mw

©2
& pwwnd

tn
Po bkRP we O OO CON A

"214 365
"192 389
"170 413
"148 323
"126115

"103 728
"081 037
"058 030
"034 697
"O10 961

2°986 804

"962 273
937 349
*912 080
"886 312

98s 481
"981 652
"981 692
"981 645
"981 551
"981 325
"981051
"980 593
"980 038

"979 335

(978 752

"978 479
978 291
978 240

‘978 380
"978 573
978 627
‘978 732
"978 840

"978 951
"978 965
"978 927
"978 942
"978 852
978 600
"978 450
"978 294
"978 134
"978 080

"978 024
"978 024
977 910
977 792
"977 613
"977 309
"976 993
"976 667
"976 264
"975 843

"189 107
"207 626
"225 974
"244 282
‘262 637
"281 086
299 761
°318 710
338 117

"358 079
‘378 744
"399 992
"421 513
"443 222
"464 982
°486 602
"508 029
"529 402
"550 670

°571 830
"592 879
"613 914
"634 987
656 045

"677 193
"698 593
"720 143
"741 849
"763 715
"785 635
807 611
"829 587
"851 677
873 885
"896 272
"918 963
"941.970
"905 303
"989 039

975 46913°013 196

"975 076
974731
"974 232
973775

"037 727
"062 651
"087 920
"113 688

018 519
018 348
"018 308
"O18 355

018 449

018 949
"O19 407
"019 962
"020 665
"021 248
"O21 521
"021 709
"021 760

"021 620
"O21 427
"021 373
"021 268
"021 160

"O21 049
"021035
"021073
7021 058
‘021 148

"021 400
"O21 550
"021 706
"021 866
.021 920

"021 976
"021 976
022 090
"022 208
022 387
‘022 691
"023 007
"023 333
"023 736
024 157
"024 531
"024 924
025 269

"026 225

10 13°829 667/1'981 226/4°170 333|0°018 774 |5°123 152] 978 307

"101 4591978 144
"079 603/977 961
"057 5641 977 766
"035 3301977 562

"012 892/977 349
"018 675 |4°990 2411977 136
"997 377] 976 923
"944 300) 976 721
"921 021/976 538

897 559) 976 383
"873 942/976 249
*850 1911 976 124
"826 315) 976 oo!
"802 316/975 874

*778 1901975 729
753 9191975 567
"729 486] 975 388
"704 874/975 192
"680 066) 974 977

"655 9431974 740
"629 783) 974 486
"604 269} 974 216
"578 485) 973 924
"552 409] 973 618

"526 027/973 305
"499 3321972975
"472 307/972 633
"444 940] 972 272
"417 212/971 886

389 098) 971 472
"300 570) 971 025
"331 595) 97° 545
"302 1401970 032
"272 1721969 485
‘241 6571 G68 gio
"210 567) 968 305
178872) 967 665
"146 537| 966 993
"113 530] 966 286

.079 816] 965 536
"045 352/964 740
010 092} 963 883
025 768 |3°973 9751 962 974
"936 949] 962 000

= mq

)
Mm POH O DO OT An SO WwW WO

we

&9
© G7 An S2WwW Ww O OO O~ OA

Go

cn pwwyHs

tn
A&® DO CO COr~? Qn

135

HM(5),
4 PERCENT.
Logarithms and Co-loga arithms of D,, N,, and M,; with

tr Differences.

2 | Colog Nz A A Colog M,

=

10 |6°876 848 991 797 4 857 405

021 693 |3°142 595

1 | *898 541

15

we

i)

©

©
7 pwend \O CONT Qn PO vm O ©) C~ AM FW Bw O WO Or A

~*~
hROb eH OC HO Cn An

_*987 108
5:009 759
"032 623
"055 700
078 979
"102 441
"126058
"149 809
"173 685
"197 684
°221 810
"246 o81
"270 514
"295 126
"319 934

"344957
“3 7O 2 17
"395 731
‘421 515
447 591

"473 973
"500 668
"527 093
*55.5 060
“582 788
"610 902
"639 430
668 405
*697 860
727 828

"7.58 343
"789 433
"821 128
"853 463
"886 470
"920 184
"954 648
"989 908
4°026 025
063 051

021 856

7 | 022 039

022 234
022 438

022 651
022 864
023 077
023 279
023 462
023 617
023 751
023 876

023 999
024 126

024271
024 433
024 612
024 808
025 023
025 260
025 514
025 784
026 076
026 382

026 695
027 025
027 367
027 728
028 114

028 528
028 975
029 455
029 968
030 515
031 090
031 695
032 335
033 007
033 714
034 464
035 260
036117
037 026
038 000

"134 392
"127 §70
"121 698
"116 167

"ITO 595
"104 805
098 275
"090 861
"081 906

"O71 104
"057 903
"O42 O54
"027 189
‘010.975

2°994 793

979 301
964 680
"950 470
936 835

"923 788
“OIT 335
"899 171
"887 145
875 410
*863 670
851 478
"839 128
"826 621
"813.957

"Bol 422
"789 018
776 887
764.755
752 625

"740 363
°727 702
"714 640
"701172
687 163
672 605
"657 614
642 179
626 418
"610 062

993 178

994 128
994 469
994 428

994 210
993 470
992 586
GOI 045
989 198
986 859
984 991
984 235
983 786
983 818

984 508
985 379
985 790
986 365
986 953
987 547
987 836

865 608

*872 430
878 302
883 833

"B89 405
B95 195
"GOT 725
"909 139
"918 094
"928 896
"942 034
"957 04
972 811
"989 025
3°005 207
"020 699
"035 320
"049 530
"063 165
7076 212
088 665
"100 829
"112 855

"124 590
"136 330
"148 522
°160 872

"173 379
*186 043

"198 578
"210982
"223 113
"235 245
"247 375
"259 037
"272 208
"285 360
298 828
"312 837

"327 395
"342 386
357 821
"373 582
"389 938

O14 621
O14 210
013 635
O13 O47

O12 453

O12 164
O12 026

O1l 735
OIT 740

O12 192
012 350
O12 507
O12 664
O12 535
O12 404
O12 131
O12 132
O12 130
O12 262

o12 661
013 062
013 468
014 009
014 558

014991

.~ Go Go iS) eo bt
0 OO OX AN FH YF OO OO ONAN BOB mM OC WOO CGOr~ T? Qn PO KH =| OC

GO YO

aN

Po Ke O WO Cr? AN

136

H™(5),
4 PERCENT.
arithms and Co-logarithms of D,, Nz, and M,; with
“ their Diferoneee —eontineay

Colog D, | Log Nz A

A A
Log Ds | (Logupz) (Colog eps)

tn

pw weno © DA

“vn

~

~r

5
6
7
8
gy
°
I
2
3
4
5
6
7
8
9

"972 636) °
971 938
971 268
"97° 472

"969 453
"968 431
"967 231
"965 916
"964 463

"963 146
"961 679
"960 252
"958 652
"956 963
"954 782
"952 122
"948 849
"945 £09
"940 942

"937 453
"933 261

"194.054
°222 116
"250 848
"280 376
"310923
"342 492
*375 261
“499 345
"444 882
"481 736
"520057

"559 805
"601 153

"644 190
"689 408
"737 286
‘788 437
"843 328
"902 386
"964 933

929 671|2°031 672

"925 159

"920 026

"102 OO!
"176 842

"O14 542| °256 816
"907 O10) “342 274
"900 830] °434 364
"893 713] “533 525
"886 583] °639 812

"B80 539) °753 229
875 089] *872 690
*867 883] ‘997 601
"861 724/1°129 718
852 265] 267 994
"838 823] °415 729
"816 019] *576.906
"789 570] °760 887
“731 Q11
662 632/0°239 406

"543 633] 579774

"O71 317

1°033 I41

"859 909'959 844
"B19 7531958 655
°778 4081957 375
"735 783/956 oo

"691 78.4) 954 542
646 3261952 983
"599 399/951 322
"55° 6311949 559
"500 190} 947 687

"447 877/945 669
"393 5491943 489
"337 035/941 108
'278 143/938 500
216 6431935 622

"152 265) 932 479
"084 744) 929 087
7013 8311925 485
2°939 316/92) 712
°861 028) 917 810

-778 838] 913 603
"692 441/909 155
‘601 596/904 239
"505 835) 898 919
"404 754/893 214

"297 968) 887 075
"185 043] 880 697
"065 740] 874 126
1°939 866} 867 336
"807 202! 860 249

667 451|852 291
"519 742) 842 704
°362 446] 831 238
"193 684) 815977
"009 6611795 532
0°805 193] 767 260
"572 453] 729 O42
301 495| 668 790
"970 285] 583 162

1°553 447/413 412
2°966 859

2°860 087/1°973 223 3°139 913/0°026 77 3°898 949 960 960] 55

166 G90} °027 364
"028 062
"028 732
029 528

"030 547
"036 569
"032 769
"034 084
"035 537
7036854
038 321
"039 748
"041 348
"043 037
7045 218
"047 878
"OSI ISI
"054 891
"059 058
"002 547
"066 739
070 329
"074 841
"079 974

"085 458
"092 ogo
"099 161
"106 287
"£13 4.17
"119 461
"124911
"132117
"138 276
"147 735
"161 177
"183 981
"210 430
268 089
"337 368

"456 367

& W DN os

“~S “A
ww

~)
0 OO Gm An FOH HO WO ON A

co

Cc
© COn~NI At fp © PD

\o
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Am OO PD

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wa

DO pow an Bd ODIO

~“ ~!
© @O~ An PO HHO WO On A

Oo oo Cc
© Or~ 7 Qi - W WY HO

\o
An Pw Ww HO

Logarithms and Co-

Colog N,

4° LOT O51
"140 O91
"180 247
"221 502
"264 217
*308 216
"353 974
"400 O91
"449 309
"499 810

°552 123
"606 454
"662 965
"721 857
°783 357

847 735
"915 256
"986 169
3°060 684
"138972
"221 162
"397 559
"398 404
"494 105
"595 240
"702 032
"814957
"934 260
2°060 134
"192 798
"332 549
"480 258
"637 554
"806 316
"99° 339

1194 807
"427 547
"698 505
"029 715

0°446 553

1°033 141

137

H™M(5),
4 PERCENT.

their Differences—(continued).

A A

039 O40 [2°53 220
040 156] °575 745
OFF 345) “557 009
042 625| "538652
043 999| °518.959

O45 458) “498 357
O47 017| °476 522
048 678| °453 499
050 441] 429955
052 313| 403 064

054 331) °375 372
"346 234
"315 486
"283 243
"249 316
"213 646
175 628
"134 693
"090 116
"O44! 372

"987 997
"930929
"869 367
"B04 137
"734 195
658 879
"577 858
"489 854
"394.429
"291 586

"181 379
"065 127
0°943 622
"815 036
‘680 735

"537 340
270958] *380 840
331 210| ‘201 546
416 838 |1°996 192
586 588) °732 643

"400 075
2°9.49 826

981 864
981 043
980 307
979 398
978 165
979977
975 556
974 009
972 308
970° 862
969 2.52
967 757
966 073
964 330
961 982
959 065
955 423
951 256
946 625

942 932
938 438
934 77°
930058
924 684

918 979
911 996
904 575
897 157
889 793
883 748
878 495
871414
865 699
856 605
843 500
820 706
794 646
736451
667 432

549 75!

|

232 740

Colog M,

982 525 |3'406 780

"424 255
"442 391
"461 348
“481 O41

"501 643
"523 478
"546 501
"579 945
"596 936
"624 628
"653 766
"684 514
"716757
"750 684

"786 354
"824 372
"865 307
"909 884
958 628
2°02 003
"069 071
°130 633
"195 863
"265 805
"341 121
"422 142
"510 146
"605 571
"708 414
818 621
"934 873
1056 378
"184 964
"319 265
462 660
"619 160
‘798 454
0°003 808

267 357

599 925
1050 174

arithms of D,, N,, and M,; with

A

O17 475
018 136
018 957
O19 693
020 602
021 835
023 023
024 444
025991
027 692
029 138
030 748
032 243
033 927
035670
038 018
040 935
044 577
048 744
053 375
057 068
061 562
065 230
069 942
075 316
o81 021
088 004
095 425
102 843

‘110 207

116 252
121 505
128 586
134 30!
143 395
156 500
179 294
205 354
263 549
332 568

450 249

OO on~

~T “A (o,)
O OWI OAL POKHO OBYVAN AWHHO

co

oe)
Crtn FOO VY = CO

Oo
PpOonrnH OO OO O™

\O
nw

we & »
ODI AR FO BHO OOVOAN FOEHO OMAN FOND

-
r-)

FR awon

uw
ROBO © Or OK

19°655 5
19°5239
19°3744
19°2105
19°0376
18°859 4
18°677 9
18°498 6
18°323 6
181609
18°0152
17°893 1
17°7903
17°6941
176010
1775054
17°3989
17°2789
171505
17°011 2
16'860 6
16°698 0
16°5266
16°348 3
16°160 5
15°9669
15°773 4
15°5758
15°3741
1571680
14°953 2
14°729 3
14493 8
142501
13°9977
13°7382
13°4750
13°208 t
12°9372
12°6639
12°388 3
12°108 2
11823 4
1175317
112367

“205 555)"
+210 620 | *
"216 370|"
+222 675"
*229 324 |
"236 174|*
"243 160] "
"250 055 |"
+2'56 786)
263 040] *

+268 647 |
"273 342|*
"277 297 |"
+280 997 | *
°284 575 |"
288 254] °
"292 350]
"296 967 | «
"301 906 |"
"307 259“
"313 053 |"
“B19 311 |"
“325 901)“
"332 761!"
"339 981 |
"347 427|"
"354871 |"
“362 469 |
"370 228 |
378.157 |"
*386 417 |“
"395 029 |“
*404 086 |"
“413 458 |
423 165 |
"433 147 |"
“443 207 |"
“453 534"
“403 954"
"474 462 |"

“485 067 |
"495 839 |"

138

HM(5),

4 PERCENT.
Values of Annutties, and Single and Annual Premiums

¢ “aaa of a Unit. Jr

a
© wD AH

wy
3 PQ pow d

WO ONAT Aw PHD 10 Or OK

© 2 2
WO ONAN pw wHO

PEKRnd

ak

10°936 1
10°31 6
10°323 0
10012

9°696 9

9°379 1
9'062 5
87458
8-431 3
81196
78119
75038
T1960
68857
65735
62583
519450
56379
5'3426
5062 4
47998
45433
4298 0
40536
38159
3°5874
3°367 6
3°163 0
29743
27991
26344
2°468 5
2291
21057
1°895 2
1663 1
1410 5
T1545

8742

6207

“3497

+000 0

540920
"552 631
“504499
"379.459
+588 581
*600 B04
612979
"625 160
“637 259
*649 246
*661 080
*672.930
*684, 767
696 702
*708 710
“720 835
*732 885
“744,696
°756 054
*766 832

“776932
+786 796
“796230
“B05 634
“814774
823 560
+832 016
“839 882
*847 138
+853 882
860217
+866 596
*873 420
"880 550
“888 646
897 572
“907 290
“917 134
"927.910
"937 668
"948 088
"961 538

btd
5 318

47 SUT
049 854
+052 348
"055 024
+057 886
060 917
064.147
7067 569
‘O71 192
°075 021
*079 133
*083 549
+088 350
*093 578

"099 312
"105 527
"112 189
“119 202
"126.490

"133 958
“147935
"150 289
“159 419
+169 184,
“179 525
“190 498
“201 749
"213 152
224 76
+236 688
+249 848
"205 387
*283 524
+306 938

"337 038
“376 398
“425 690
“495 104
“578 573
"702.470
“961 538

ee
SE

TWO LIVES.

|

H™.

THREE PER-CENT.

140

H™,
3 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Inives. Equal Ages.

z Griz 24,—G,.5 2 Gz x
10 21°0079 27°2889 55 8°8676 15°3200
I 8518 1388 6 "5303 149181
2 "6497 26-9787 7 "1945 5121
3 "4111 "8095 8 7°8604 -1022
4 "1465 6329 9 "5276 13°6890
15 19°8661 4501 60 "1988 2730
6 "5801 2629 I 6°87 53 12°8557
7 "2991 0731 2 "5581 "4375
8 "0334 25-8820 3 2482 0192
9 18°7936 *6922 4 5°9468 11-6014
20 "5817 5033 65 6519 "1839
I "3838 3130 6 "3621 10-7661
2 "1926 1198 7 "0757 3479
3 17°9975 24°9217 8 4°79 14 9-9290
4 7908 ‘7170 9 "5057 5089
25 5703 5055 70 "2226 0904
6 "3398 "2874 I 3°9453 86765
7 "IOrl 0635 2 "6788 2708
8 16°8600 23°8344 3 "4279 78773
9 6176 "6004 4 "1975 4989
30 "3734 3614 75 2°9876 1352
I "1281 "1173 6 4852 67788
2 15°8782 22-8670 ” "5024 "4312
3 6229 *6109 8 "4068 0914
4 "3615 3483 9 "2247 5:7585
35 "0950 0792 80 0488 -4360
6 14°8242 21-8040 I 18844 1288
7 "5509 5231 2 "7379 4°8415
8 "2752 2360 3 "6069 ‘6711
9 "0954 20-9424 4 "4970 3190
40 13°7103 "6421 85 "4025 0751
I °4173 3341 6 °3139 3°8269
2 "L142 0182 7 *2224 0630
3 12°8024 19-6942 8 "1305 2809
4 "4848 3634 9 "0122 2-9610
45 "1619 ‘0251 go 0°8693 6105
6 | 11°8379 18-6813 I "7246 "2498
7 "5148 -3320 2 "5800 1:8776
8 "1921 17-9773 3 "4198 4826
9 10°8681 6165 4 2668 ‘0872
50 5428 -2498 95 "1321 0:6983
I "2140 16-8764 6 0328 3238
2 98802 "4960
3 "5434 1094
4 "2058 | 15-7172

° H™,
3 PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lnves.

20°9287 | 20°8243 | 20°6987 | 20°5572 | 20°4046

27°2150 "7494| °6264| °4873| °3371
1383 | 27-0601 "5290| °3926] ‘2451
0600 | 26-9792 | 26-8955 "2773 | 1327
26:9809 | 8977) ‘8113 | 26-7227 "0047

9019 8163 "7272 6357 | 26°5431

*8239 ‘7358 6440 5496 | -4540 | 26°3582
‘7476 "6571 5626 4655 3668 "2678
6733 5806 4835 3835 °2820| -1800
6021 5070 ‘4076 3048 | +2004 0955

6334 "4361 3342 2290 "1218 0140
4666 "3672 2629 "1551 0453 | 25-9347
"4011 2995 1929 0825 | 25-9700 | -8568
3362 2325 1235 ‘0107 "8955 "7794
2715 1656 ‘0543 | 25-9389 8211 7022

2069 0991 | 25:9854 "8674 “7469 6252
1427 0328 9169 7964 6731 5485
0791 | 25-9672 8490 ‘7260| -6000| °4726
0166 | °9025 7821 6567 "5281 3978
25°9548 | -8390| ‘7162 5884 4571 3242

8941 ‘7763 | °6516 ‘5211 3874 | °2517
8343 “T7147 5878 4552 | :3187 1804
‘T7752 6539 0248 3898 | 2511 1099
7171 5940 4629 3256 "1843 ‘0407
6597 5349 ‘4018 2623 | °1184 | 24-9722

6030 4765 3415 1997 "0535 9047
5471 4191 2821 1382 | 24:9896 | -8383
4923 | °3627 ‘2239 "0779 9270) ‘7731
4384 "3073 1666 0186 *8654| ‘7091
3854 2528 1103 | 24-9602 8048 6462

3333 | +1992 0550} -9029| -7452 5842
2819 1465 0005 8464 | -6866 6232
2314] -0946 | 24-9470 ‘7908 6288} °4632
1815 0434 | °8942 ‘7362 | 5720) -4040
1328 | 24-9934 *8426| -6826 5164) 3461

‘0849| -9444; -7919| -6301 4618 | .2893
0385 8967 7427 ‘5791 4087 | :2341
24°9932 "8502 6947 5293 | -3569 "1802
9492 "8051 6482 "4811 ‘3067 "1279
9064] -7612 6029 4340] :2578| 0769

‘8647 | 7187 5589] 3883] 2102; 0273
8243) ‘6771 ‘5161 3439 1639 | 23-9790
7850 | -6369 4744] -3007| °1189 ‘9320
“7469 ‘5979 4342| +2588) 0752) -8864
7100} 5602} -3952| -2183| -0329/ -8422

ld
oo

ww bh
&9 we e oe
© Or Aw SO Hm O OO Orr an PHO NHeH GO WO G1 Qn HPO HN =| O

&

&o
Ls)

fe)
I
2
3
4
5
6
7
8
9
°
I
2
3
4
25
6
7
8
9
fe)
I
2
3
4
5
6
7
8
9

fb
r°]
aN
co]

= -S
& OC VY my

wi
hw Ye O WO CO™ Qn PO WH &

wr
h&bwweO OO CONTA

" oe . Joint Life Annuity, at Top.
[156) Age of elder Life:— Last Survivor Annuity, at Side.

i) b=

we

Ge &o
5) OO Cnr QAM #F0 Bw O WO CONF AN POOH Sw CO WO DO An HPO HD

oe tm & YD my

wr
hone O OO COr~T AN

[157]

142

H™,
3 PER-CENT.

Values of Annuities on Two Joint Inves, and on the Last Survivor
of Two Lnves.

20°2460 | 20°0869 19 9328

"1810
"0917
19°9822
"8572
°7214

26°1697
0786
25°9912

9068
"8246
7438
6636
5835

5036
4241
3452
"2675
"1910

1157
0416
24°9684
8963
"8252

"7550
6859
6183
"5517
4862

4219
3585
2960
2345
1743

"151
‘0577
0016
23°9472
8940

8424
7920
7430
6954
6492

"0243
19°9377
*8309
"7090

"5704
"4379

25-9793
“8886

8014
7163
6325
5495
-£665

3836
“3011
2193
1386
0591

24:9809
9040
8279
7531
6792

6064
5346
4642
"3951
3271

2602
"1945
1296
0656
0031

23°9416
‘8819
8237
‘7671
‘7119

6581
6057
5547
5052
4570

Age of elder Life :—

8724
"7884
"6845
"5054
"4358
*3006
"1645

25°7887

6984
6104
5238
"4278
3518

2660
"1805
0957
0120
24°9296

"8485
-7687
6898
6122
5355

4599
3855
3125
2408
1703

1010
0326
23°9655
8992
"8343

‘7706
7086
"6483
5896
5323

4767
4223
3694
3180
"2681

19°7892
"7312
"6495
"5484
‘4322

"3955
"1732
"0404
189119

25-5993
5083
-4188
3300
"2410

1522
0637
24°9759
8893
8040

“7200
6373
0556
"4751
3957

3174
2403
"1649
0906
0176

23°9459
8753
8057
7372
6701

6043
5403
‘4778
"4172
"3581

3006
2445
1899
1368
0853

19°6575
‘6017
"5225
"4238
"BZ 104
"1866
"0572
189272
"8018
"6861

25-4097
3172
2255
1337

0419
24°9505
8598
‘7701
6819

*5950
5095
4249
3418
-2596

"1786
0990
0208
23°9442
8688

"7947
‘7218
6500
5793
5102

"4424
"3764
3121
2496
1888

"1296
0719
0158
22°9612
9082

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

195 302
"4705
"3997
"3036
"1928
70718
18°9453
"8182
"6957
"5830
4812

25°2179
123)
0284

24:9337
*8392
7455
6530
5617

‘4719
3835
2961
2100
"1251

0414
23°9591
8785
"7992
7214

"6449
5697
4957
"4228
3515

2817
2138
1477
0834
0209

22°9600
9009
8432
7872
7329

aN Gs & re) » «
O OWT A PWOWKH HO OHO OYA F&O HHO O OTA PO HHO

dH Ye O

J.

POkw HO ©) CO~ AN

i) = a!
OO c~r An $Ww BY» O

> b wo wo re)
hwo me OC OO W~ 7? An PDH w Oo On™ AN S$ & Be EO OO OT AH PO YD mm OC

tA

[158]

19°403 5

"3520
"2775
"1840
"0759

18°9575
"8339
7098
"5901
"4803
"3815
°2867

25-0224
24°9245

8268
7292
"6324
5368
4426

"3497
2583
"1680
0790
23°9912

9047
8196
7363
6545
5740

"4952
"4175
3412
‘2661
1927

1207
0509
22°9829
9168
8525

‘7900
‘7291
6700
6126
*5570

192718

"2224
"1503
"0592
18°9538
"8383
“7175
"5962
"4795
"3725
"2706
"1849
"0934

24:8210

7200
6193
5194
4205
3231

2272
"1326
"0392
23°9473
8565

‘7670
6790
5928
5082
"4252

3436
2634
"1846
1071
0314

22-9572
8852
8152
7472
6810

6168
5542
4934
4346
OT74

Age of elder Life :—

143.

H™,
3 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lnves.

19°1308

0836
0138
18°9253
"8225

"7098
"5919
"4735
3598
"2558
°1627
"0739
17°9856
"8925

24°6130
5090
4057
3037
2029

1036
0058
23°9091
8140
‘7200

"6274
5363
447)
3595
‘2736

1892
1062
0248
22°9447
8664

‘7898
7155
6432
5731
5049

4387
3743
‘3117
2511
"1924

18°9794

"9341
"8667
*7808
"6807

5708
"4558
"3404
"2296
*1286

"0385
17°9526
8673
7775
6788

24°3983
2915
1860
0819

23°9792
‘8779
‘7778
6792
5819

-4860
3916
2991
2083
1193

‘0319
22°9460
8615
7787
6977

6184
5416
4668
3944
3239

2555
"1891
1245
0620
0014

18°8193

"7761
"7 109
6275
°5302
"4232
*3I10
"1986
"0908
17°9928
"9056
*8228
"7406
"6539
"5585

4532

24-1774
‘0682
23°9605

8543
"7494
6458
5436
"4427

3432
2454
1495
0554
22°9630

8724
7833
6958
6098
5260

"4439
3642
"2868
2118
1389

0682
21-9995
‘9328
8682
8056

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

18°6516
"6104
5475
"4666
"3720
"2678
"1586
"O4QI
17°9443
"8493
"7650
"6852
*6061
"5225
"43095
"3287
"2185

23°9509
8394

‘7294
6210
6136
“4077
3031

2000
0985
22°9991
‘9014
8055

‘7114
6190
5282
“4390
3520

2669
1843
1041
0263
21:9508

8776
8064
7374
6705
6059

> wo ©o we re) = bos

-— GO & mM

p>

wa
&Owds QO WO O~ An

HX,
3 PER-CENT.
Values of Annusties on Two Joint [nves, and on the Last Survivor
of Two Laves.
28 29 | 30 31 32 33
ro | 18°4790 | 18°3026 | 18°1217 | 17°9368 | 17° 7458 17°5482 | 10
I "44.00 "2653 "0864 "9033 "7140 *5182 |] 1
2 *3793 °2070 "0300 "8491 *6620 "4682 | 2
3 | °3008) ‘1309, 17°9566) °7778) °5931| 4016] 3
4 2088! ‘o416/ ‘8697 6937 "5112 "3223 |] 4
15 | °1075]17°9429| °7738) ‘6004] °4208| 2343 | 15
6} ‘oorz| °8395| °6732}] ‘5026| °3257| ‘1421 ] 6
7 | 17°8947 7360} °5726| ‘4048 *2308 "0499 | 7
8 "7929| °6371 "$766] “3117 "1405 | 16°9624 | 8
9 | °7008) °5479| °3903| °2283) “o599| 8847; 9
20 *6196| °4696| °3149] °1557] 16°9902 "8176 | 20
1} 5426} °3957| °2439| ‘0876/ ‘9249) °7553| 1
2 "4666| °3226| °1739| ‘0206; -8608 “6941 | 2
3} 3863} ‘2455} 0998] 16°9497| °7930| °6292] 3
4| ‘2974| ‘1600|} ‘0177| °8708| °7174| °5568) 4
25 "I9QI | 0650] 169261 | °7827) °6327| °4756| 25
6 | °0926| 169621 | °8267] °6869] *5404] °3869] 6
7 | 16°9786 "B5IQ}| °7203 *5840} °4413 "29015 | 7
8 7367 | “*6ogo) °4766] °3375| ‘1915| 8
9 | 23°7195 49341] °3049| °2299| 0877) 9
30 6056 | 23-4830 °2486] °1176| 15°9796 | 30
I 4933} 3668 | 23:2415 ‘oor0 | «8670 | 1
2 *3823| -2517| :1224 | 22-9943 7483 | 2
3 ‘2726; °1882| -0047| -8726 | 22-7412 3
4 1641 0258 | 22°8884| °7520| °6164| 22:4819| 4
35 0571 | 22°9148 | +7735) 6330} -4930 3541 | 35
6 | 22°9518| ‘8057| -6601 5156 | :3713 2280 | 6
1] 8487 | 6986, -5491 4002] :2518 1040 | 7
8 7473| °5934) °4398| -2870| +1341] 21:9820| 8
9 6478 | 4902) 3327] :1756| -0186 8619 | 9
40 ‘5502| +3887) :2274] -0663 | 21-9050 "7440 | 40
I 4541 2891 1239 | 21:9588 | °7933 “6279 | 1
2 °3599} 1912) -0222| -8531 "6835 6137) 2
3 3674! ‘0951 | 21-9223) 7494] -5756 4016] 3
4 1770| ‘0013; :8249] ‘6481 "4704 2921) 4-
45 ‘0887 | 21:°9095 | -7296| ‘5491 3674 "1851 | 45
6 0030| ‘8206; -6372| °4531 2677 0811 | 6
7121-9197} 7342] -5475| °3599) +1707) 20-9804 | 7
8 8391 6506 | °4607| -2697 ‘0769 8828 | 8
9 ‘7608 | ‘5693| °3764} +1821 | 20-9859 7881 | 9
5° 6849} °4906| -2947| 0973] -8977 6963 | 50
I 6113} °4142) 2155] -0150; :8123 6075 | 1
2 5397 | °3401 "1386 | 20°9353 | -7295 5213 | 2
3 4705 | °-2684|) 0643) -8582| -6494 "4381 | 3
4 4036} -1991 | 20:9924| -7838| ‘5721 3578 | 4

[159]

Age of elder Life: —

144

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

10

|
p © WD &

YO cn~r Qn

a

re)

Go

°
I
2
3
4
5
6
7
8
9

30
I
2
3
4
5
6
7
8
9

- - GO b oe)

ja
-&OoO He CO WO Or Qn

[160]

34

17°3436

°3153
"2673
°2029
°1262

"0408
16°9512
"8618
“777%
"7021
6378
"5782
"5199
"4.580
*3888

"3109
"2258
"1341
"0380
15°9381
‘8339
"7256
‘OIII
"4899

22°2161
0855
21-9568
8303
7057

5830
4625
3437
2271
"1132

0017
20°8937
"7886
6871
"5884

4930
4004
3107
"2241
"1406

35

17°1325

"1059
0508
16°9977
"9233

"B4O5
"7536
6668
"5849
"5126

"4510
"3941
°3386
°2797
"2136

"1390
0575
15°9694
"8772
"7813
"6810
"5768
"4667
°3499
"2259

21°9441
8108
6794
5501

4228
2974
1740
0526
20°9341

8181
‘7054
5962
4902
"3875

2880
"1916
0981
0079
19-9209

Age of elder Life:—

145

H™.
3 PER-CENT.

Values of Annuities on Two Joint Ltves, and on the Last Survivor
of Two Inves.

36

16°91 54

"8903
"8462
"7862
"7142

"6339
°5497
"4656
"3863
"3167

"2576

"2034,

"1507
"0047
"0317
15°9604
"8823
‘7979
"7095
"6174
"5214
"4212
"3154
"2030
"0835

14°9571

21-6660 |

5299
3956

2635
1332
0048
20°8787
7552

6344
5171
4032
2929
"1857

0821
19: 9814
"8841
‘7900
6993

16°6931
"6696
6273
"5694
°4997
"4220
*3402
"2.589
°1822
“1150
"0587
"0069
15°9569
"9038
8438
7758
“FOIL
"6202
"5355
4474
°3553
"2595

1578 |

"0499
14°9351

8133
6851

21°3821
‘2429

1057
20°9704
8370
"7056
5772

4514
3293
2105
0955
19-9838

8756
7708
6691
5710
"4764

16°4656
"4436
"4032
"3473
"2799
"2046
"1254
"0466
15°9725
"9079
"8539
"8048
7573
"7970
"6500

"5852
"5138
"4365
"3555
"2712
"1832
"0913
T4'9941
"8905
"7802

6633
"5398
"4105

21-0918

20-9495
°8090
6704
0339
4001

2693
"1419
0182
19:8983
‘7818

-6690
5594
4535
3510
"2523

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

39

16°2319

"2114
"1728
"1190
70538
15°9808
"9042
8279
"7593
6942
"6426
"5959
“5511
"5933
"4492

"3875
"3195
"2457
"1683
"0877
"0036
14°9160
"8229
"7239
"6181

5°59
"3874
*2630
°1327

20-7949
"6492
5052
3632
2242

‘0879
19-9553
8263
7013
5798

4620
3477
"2369
"1300
0268

= -
Oo con~ Qn $+ © LH HO

dS

we

&o

wn Fa aad +. Go
0 OO IAN FPWOH HO OH ONAN PW HHO OO OAH POV O

—- WP &

146

H™,
3 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

IO | 15°9913
"9723
"9354
*8836
"8207

"7501
6758
"6021
"5329
"4732
"4240
"3797
"3372
"2922
"24009
"1822
"1174
"0471
14 313"
"8965
"8162
"7326
6438
"5491
"4481
"3405
"2268
"1075
13°9823
"8502

bug
Oo Ou Arn PO HD =

)
o

P& wo 4

be

>
WO wor An PHO wHeEH GO HO CHAN

Go

as
co)

20-4909
3414
1939
0493

19-9075
‘7693
°6350
5045
3777

2548
1354
0198
189079
8002

> GO HN -m

p>

hOoObH =m O WO COnm~T AN

Cr

[161]

4!

15°7422
7245
"6894
"6396
"5788
"5106
"4387
"3673
*3008
"2433

"1964
“1544
"1144
"0719
"0234
14°9676
"9060
"8390
*7688
"6956
"6192
"5396
“455°
"3647
2681

"1654
"0566
13°9423
"8223
"0954
"5610

20°1790
0259
19-8755

‘7279
5841
"4440
"3079
‘1756

"0472
18-9225
8016
6847
5718

Age of elder Life :—

15°48 32

"4669

"4334
"3857
"3271
"2611
“IQT7
"1227
"0584
°0034

14°9587
"9189
"8812
"8412
"7953

"7426
"6840
"6203
"5535
*4840

"4114
"3358

1.116
18-9735

8393
‘7090
5825
4602
3421

43

"2002
°1683

"1224
"0660

"0024
14°93 53
"8688
"8068
°7 540

WALK
"6739
"6384
*6008
"59575

°5°75
"4521
3916
°3281
"2622

"1934
°1216
°04 53
13°9636
"8761

"7828
"6837
°5797
°4700
"3540
"2306
"0981
12°9553

19:5319

3722
2162
‘0641
18:9161
‘7719
6318
"4956
3632
"2352
"1115 | 17-8813

44

19-1974
0352
18-8768
"7225
5721

"4257
2833
"1449
0108

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

15°2152 | 14°9397 *4°6570
"9260
"8957
"8518
“1974

"7301
"6713
"6071
"5475
"4969

"4504
*4210
"3876
3523
"3116

"2643
"2117
"1544

0043
0318

13°9666
"8987
"8268
"7489
6658

‘5770
*4830
"3839
*2796
"1688
"O510
12°9243
"7874
6405

6444

6158
"5737
“$214

"4623
"3999
"3380
*2806
"2321

"1936
*1602
"1290
0959
0576
"0130
13°9632
"9089
"8520
"7930
7313
6671
"5987
°§253
"4467
3625
"2732
"1791
"0798
12°9745
"8622
7413
"6104
"4696
"3202

18-8565
6917
5309
"3740

2213
0724
17-9277
‘7874
"6517

i=) ome a]
~O YN & O

POP we O WO C™ AN

© e
OO COs Qn

Go
Oo corr Qn SOB = O

pp

wr
Pwee O OO CON A

RR pwee

Oo

147

H™,
8 PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lnves.

Ln |

WO Oran POKHO

14°3695 | 14°0786 | 13°7839 | 13°4843 13" 1800 | 12° Bene
3582} ‘o685| ° " 8634
3311) "oq2g] ° 45. "8433
"2908 ‘oog4 | *7T: ’ “18: "8116
"2406 {13°9562| ° ° ° "7719

1836] ‘g013| ° 3235) ° "7243
1234] °8433] 5. 12°9754| °6747
0638| -7858| . 9243| °6256
"0087 "7328 “4! ° "8773 "5806
13°96221 ‘6885| - ° "8386) °5436

9257! °6538| ° ‘Bog2| = °5158
8942 | “6aqr] ° "7847 | °4927
8649 "5967 . ‘ "762.5 "4723
8340} °5678| ° 7391 | *4506
7980] °5341 . ’ “FITS "4248

"75591 °4945) * "6787 "3940
77090} = =*4502 ° ; "6417 "3593
6576] <“4016| ° "87: "6010 "3211
"6038! *3509|] ° ; "5586 | ‘2811
*§480) ‘2982; ° ° "5147 "2400
"4898 | 9 °2.433 | 12° 4690) "1971
"429021 ‘'3862) -g: ° "4217 "1529
3045} "1953| ° "3712 | = “1055
2954) "0599/ ° "3169 } °0546
2208 | 12°9897| ° . "2.582 | 11°9997

“I413 | "9143, * " 1954] 9407
0566) °8343| 60: "1283 | °8779
12°9673 °7499 “4 "29! "0577 "8114
‘8733 | “6608) - 119829] ‘7414
°7732| °5660| ° , 9032] °6664
"6665; °4646| ° . "8177 "5860
"5512 ) °355T] * ; 7248} °4984
"4263 *2360 039. 83: 6232 "4024
"2917 "1076 . . "5128 "2979
"1485 | 11°9707| ° "3947| °1860
11°9966] ‘B2a52| ° . "2685 0663
"6730| ° "3: "1361 | 10°9403

18-5100 . ° 10°9979 | ‘8089
3427 | 18-1580 ’ 8535 "6714
1792 | 17:9878 | 17-8004 "7020| °5268

0198 "8218 6275 | 17-4366 3748
17°8645 6597 4585 | -2607 | 17:0667

7132 5017 2936} -0887 | 16-8876 | 16-6899

5663 3482 "1333 | 169214] 7130 5080

4243 1995 | 16-9779 "7590 5434 3310

Oo 8 = O

oo]

w

we
~&OO Nm GQ WO COr~r Qn

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Oo
Oo ONAN HPWH OO OO GO Ah

so
Go

°
I
2
3
4
5
6
7
8
9
30
I
2
3
4
5
6
7
8
9

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O orAh AO nd
in
pw end OTA BO BDHO

te Joint Life Annuity, at Top.
[162] Age of elder Life:— } 5 ost Survivor Annuity, at Side.

148

H™,
3 PER-CENT,

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

53 53 5+ 55 56

baa

£2°§515 | 12°2279 | 11° $999 It ‘5676 11°2321
"§465| °2238| °8966) 5652] °2305
"5279 | ‘2064| °8805} °5503| °2168
4977] 1779] °8535| 5249] "1929
4589} ‘1409}] °8183] ‘4914] ‘I6II
4142} °0981| °7774) °4524| °1239
*3666) °0525 7338 *4.106 "0841
"3195 | °0073} 6906} °3692] "0445
"2764 | 11°Q661 "6511 *3316| °0085
"2411 "9325 6191 *3032 | 10°9797

2148} °9077| 5958] °2791| "9590
"1933 | °8876| °5770| °2617| °9427
1743 | ‘8700} °5607) °2467} ‘9290
1543 | °8514] 5437] ‘2310] "9145
"1303 | °8292] ‘5230| ‘2119] ‘8969
"1015 | °8023 4980} °1886] 8753
70689} °*7718| °4695| ‘1620} ‘8504
0330) °7382| °4379) °1325| *8229
11°9956| "7031| “4@51} “10Ig}] °7943
9570| °6670} 3714) ‘o704} °7650

9169} 6295] °3365| ‘0378! °7347

od wo
mo

we e
OO ONAN F&O NK GO OO ONAN PO HDHD OO GO 13 An PO WH» O

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O DO Ann AW HHO WO ONAN AW DK CEC [OO CHr 7 AN + © &W

*B755} °59°9| 3004] ‘0043| °7036
"8312 "5496 2620 | 10°9686 "6704
"7337 "$052 "2.206 "9301 6347
"73231 °4572| °3758) -8884) “5959

"6771 | °40560)} °1277) 8436) °5542
6181 | =—°3505| 0703] 7957} 5098
"5§60| °2924| "0221) °7453] *4629
"4902 "2310 | 10°9648 *6920] ‘4134
4201} °1653] 9036] 6349] °3605
3445] °0947| 8375] °5734| °3033
°2622| ‘Or7$] °7654| “so60} °2405
"1718 | 10°9324| °6856| 4314] °1708
0732) °8395| °5983| °3495| °°943
10°9673| °7397| °5043| 2613] ‘o116
8539] °6325| °4033| °1663} 9°9225
"7345 "5197 "2968 "0660 "8283
*6098} 4016} °1854] 9°9611 "7297
"4792; °2778| °0683] 8508! ‘6259
3417] °1473| 9°9448| °7342| 5162

"1968| °0097) “8144|] ‘6109] °3999
0434} 9°8636| 6757] ‘4795| °2758
"7080| °5276/ °3390| ‘1428

16°3065 3707 "1898 "0013
1220 | 15-9172 "0328 | 88521

Go
Go

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{ Joint Life Annuity, at Top.

[163] Age of elder Life:— ( Last Survivor Annuity, at Side.

149

H™,
3 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Laves.

62

ton}

oad
O D3 An PWKHHO

Io | 10°5534 | 10°2104 9°1815
I "5533 | ‘2109 8] ° "1837
"5417 "2004 . ° "1761
5207} ‘1806] ° ’ "1603
4922} = *1538] " "1379
"4586! ‘x2I9] ° . "1109
"4224 . . . "0815
*3864 "O5. ° ° "0522
"3538 e e . "02 56
"3277 | 9 ; ; "0047

3094) : "3200 8'9907
"2955 |” "9805
2839] °9: ; "9727
"2719) * " "9945
2570] 9541
"2383 : : . "9406
2168) ° , ; "9247
"1929|] ° ; "9070
1680) ° ; ° 8885
"1427 . . . "8699
"1166| * ° ° "8509

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"0900 e e ® "8317
0616) ° . . "8112
"0308 | ° , ’ "7891
9°9976| ° "7051

&s

9618} * ‘7392
"9236 e ° ° 7 Ir 7
*8834| ° . “6828
‘B410| 547! . . "6525
7956) °s . . "6200

7465| “5849
6925] ° "5459
"6321 “3! . . "5018
"5656 ° ° ° "4529
4935] ° "3996
"4156 : . . "3416
"3331 . ° . "2801
"2467 . . . "2156
"1556 ® e ° "1476
‘Osgl ° e e °075 I

89565] ° 7°9978
8467) ° "9145
7284] * ; , "8240
*6022| ° . "7267
"4686| ° . "6230

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oe Joint Life Annuity, at Top.
(164) Age of elder Life:— Last Survivor Anntity, at Side.

150

Ho®™,
3 PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lrves.

68 69

|

fm
OO GD? An HG BO MO

78373 71706 | 68347
‘8412 | ° "1752 | °8394
"8370 | : "1723 | °8372
"8257 | - "1630 | °8289
"8087 | - "1485 | °*8155

"7876 | ° "1302 | °7985
"7642 | ° "1096 | °7793
"7408 | ° ‘0889 | "7599
"7395 | ° "0699 | °7421
"7039 | * "0554 | °7285
"6924 | ° 0463 | °7200
6852 | ° 70403 | °7146
‘680r | ° 70363 | °7IIO
"6749 | ° "0323 | *7076
"6680 | : "0267 | °7026
6585 | 0188 | -6955
“6471 | ° 70093 | °6868
"6341 | ° 6°9984 | °6768
"6208 | ° "9872 | °6666
"6074 | ° 9759 | 6563

"5938 | - "9646 | °6460
"5803 | ° "9534 | *6359
"5659 | ° 9415 | *6251
"5504 | ° "9287 | °6135
5335 | ° 9147 | *6008
5154) ° "8996 | °5871
"4960 ) ° "8836 | +5726
"4759 | ° "8671 | °5576
"4549 | ° "8499 | "5421
"4325 | ° "8315 | °5256
"4083 | ° 8117 | *5078
“3811 | ° "7894 | -4878
"3501 | ° "7639 | °4646
"3153 | ° 7349 | °4383
‘2772 "7031 | *4094

2354) ° "6682 | °3777
"1912 | ° "6312 | °3440
"1448 | "5925 | °3088
"0959 | ° "5517 | °2777
"0437 | ° "5082 | °2321
6°9878 | - 4615 | °1897
"9272 | ° "4108 | 1435
"8609 | ° "3549 | °0926
"7890 | ° "2943 | °0371
“71IQ | * "2290 | 5°9774

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Joint Life Annuity, at Top.

[165] Age of elder Life:— Last Survivor Annuity, at Side.

151

H®™,
3 PER-CENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lnves.

a]

0 COX AK & © WD me O
bt

Pw bw HO

5°5362
"5414
"5412
"5361
"5269

"5147
"5004
"4857
"4722
"4619

"4557
"4520
"4500
"4481
"4452

"4406
"4347
"4278
"42.08
4138
*4068
"4001
"3930
"3852
"3768

"3677
"3581
"3483
"3383
°3277
"3163
"3034
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"2709
"2516

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-- , _§ Joint Life Annuity, at Top.
[166] Age of elder Life: Last Survivor Annuity, at Side.

152

H™.
3 PERCENT.
Values of Annuities on Two Joint Iives, and on the Last Survivor
of Two Inves.

Ln

be pot
Oo On Ann POS DH =| O

Go

I
2
3
4
5
6
7
8
9
30
I
2
3
4
5
6
7
8
9

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$— OF DY mn

-. _‘ § Joint Life Annuity, at Top.
[167] Age of elder Life: Last Survivor Annuity, at Side.

153

H™,
3 PER-OENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

82 83 84 85 86

3°2441 | 3°0480 | 2°8711 | 2°7056 | 2°5407 | 2°3665 | 10

I | °2484 | ‘0522 | ‘8751 | ‘70905 | ‘5444] °3700] 1

2| ‘2501 | ‘0540 | °8769/] ‘7114 ] °5463 | °3719 | 2

3} ‘2492 | °0533 | °8765 | ‘71131 *5463 | ‘3721 | 3

4 | ‘2460 ) ‘0505 | ‘8741 | “7091 | °5447} 3708 | 4

15 | ‘a$09 | 0459 | °8700 | “7055 | °5414 | °3681 | 15
6 | °234$ 1} ‘ogoo | °8646 | *7006| ‘5370 ] °3642 |] 6

7 | °2272 | °0334 | °8585 | “6950 | °5319 | ‘3599| 7

8 | +2203 | *0270| ‘8525 | ‘6895 | °52369} ‘3550| 8

Qj} ‘2150 | °0220 | ‘8480 |] “6852 | ‘5229 }] °3513 | 9

20 | ‘2119 | ‘o192 | °8454 | °6828 | °5206 | °3493 | 20
I | ‘2103 | ‘0178 | *8440 | 6816] ‘5195 | °3483 1

2] ‘2099 | °0174 | °8438 | °6813 | ‘5194 ] °3481 | 2

3] *2098 | ‘0174 | ‘8438 | “6814 | “5195 | °3483 | 3

4 | ‘2091 | ‘0169 | °8433 | ‘6811 | °5193 | “3482 | 4

25 | *2076] ‘0155 | °8422 | “6801 | °5185 | °3475 | 25
6| °2054 |] °0136| °8405 | ‘6786] °5171 | °3464] 6

7 | °2026 |] ‘orrr 8382 | 6766) °5153 "3448 | 7

8} "1998 | *0085 |) °8359 | °6745 | °5135 | ‘3432 |] 8

9} °1970 | ‘0060 | °8336 | °6725 | °5117 | °3416] 9

39] 1943 | "0036 | °8315 | °*6705 | “5099 | “3407 | 30
I] ‘1918 | ‘oo1r3 |] °8295 | ‘6687 | °5084 | °3387] I

2] ‘1891 | 2°99990 | 1°8274 | ‘6669 | ‘5068 | °3373 | 2

3 | *1863 | "9964 | ‘8251 | ‘6649 |} “5050 | °3358) 3

41 °1831 | °9936| °8225 | °6626| ‘5029 | °3340] 4

35 | *1796| ‘9905 | *8197 | “6600 | +5007 | +3320 | 35
6| ‘1760 | ‘9872 | ‘8168 | °6574 | °4983 | °3300| 6

7 | '1724 | ‘9840 | “8139 | °6548 | ‘4961 | °3280] 7

8 | °1688 | ‘980g | ‘8r12 | °6524] ‘4939 | °3261 | 8

9 | °1653 | ‘9777 | °8084 | ‘6500 | ‘4919 | °3243 | 9

40 | 1616 | °9745 | ‘8056 | °6476} °4898 | ‘3226 ] 40
I} 1573 | °9707 | °8023 | °6447 | ‘4873 | °3205] 1

2| ‘1520 |] ‘966: | °*7982 | ‘6411 | °*4842 | °3179] 2

3 | ‘1456 | ‘9604 | *7932 | °6366| °4803 | °3145| 3

4} ‘1384 | "9540 |} ‘7874 | °6315 | °4758 | °3106| 4

45 | °1303 | °9467 | *7809 | °6257 | °4706 | °3061 | 45
6} 1216 | *9389 | °7739 | *6195 | ‘4651 | ‘3013 |] 6

7 | °1128 | ‘9310 | *7668 | °6132 | °4595 | °2964| 7

8 | °1036 | ‘9229 | °7596 | “6067 | “4539 | ‘2915| 8

Q| "0940 | °9143 | ‘7520 | ‘6000 } 4479 | ‘2864 {1 9

© | °0838 | ‘9052 | ‘7439 | ‘5929 | “4417 | ‘2810 | 50

1] °0726 | °8953 | °7351 | “5851 | “4349 | 2752] 1

2] ‘0598 | °8839 | °7250|] °5761 | °4270| °2684 |] 2

3| °0455 | ‘8712 | °7137 | °5661 | *4182 | ‘2607; 3

4| ‘0300 | °8573 | °7013 | ‘5551 | °4086 4

oe, Joint Life Annuity, at Top.
[168] Age of elder Life:-— ) y a4 Survivor Annuity, at Side.

154

H™.,
8 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

88 89 go

oo

& & WH HO

2°1829 | 1°9673 | 1°7242 | 1°4747
"1862 | ‘9702 ] °7267 1 °4768
"1881 | °9720 | °7283 | 4781
"1885 | 9725 | °7288 | -4787
"1875 | °9719 | °7285 | -4786

)
I

2

3

4

5 | ‘1852 | ‘g7or | °7272 | °4776
6 | ‘1819 | °9673 | °7250! *4760
; "1778 | "9039 | °7222 | *4739
9
°
I
2
3
4

ees

"1737 | °9603 | °7192 | *4714
"1704 | °9574 | °7167 | *4694

"1685 | "9557 | ‘7153 | *4682
"1676 | "9549 | °7145 | °4675
"1675 | "9548 | "7145 | °*4675
"1077 | "9551 | °7147 | 4677
"1677 | 9551 | °7148 | °4678

we
wn Owe GO OO ON AN

S

"1072 | "9547 | “7145 | *4676
"1662 | "9539 | °7139 | °4672
"1648 | °9528 | °7130 | °4665
"1634 | ‘95161 ‘7121 | °4658
"1620 | *9505 |] ‘7IIr | °4650

s)

6

/

8

9

Oo} 1607 | "9494 | °7102 | +4643
I] “1596 | 9484 | “7095 | *4638
2) °1584| 9475 | °7088 | -4632
3| "1572 | °9404} "7079 | *4626
4] *1556| ‘9452 | ‘7069 | *4618
5

6

7

8

9

O

I

2

3

4.

&o

"1539 | °9438 | °7058 | *46ro
"1521 | ‘9422 | *7046 1 74600
"1504 | °9408 | °7034) “4591
"1488 | °9395 | °7023 | °4582
"1473 | 9382 | °7014 | °4575

‘1459 | "9371 | °7005 | +4568
"1442 | °9358 | “6905 | “4561
"1420 | "9340 | *6981 | *4551
"1392 | ‘9317 | °6964 | *4538
"1359 | "9291 | °6943 | °4522

Go
CO O~3 An PbhwOHKH =m CO /O CYT OA

>

5 | °1320 | °9258]) ‘6917 ] ‘4502
6] °1279] °9224 | ‘6889 | ‘4481
7 | 1237 | *9189 | *6861 | “4459
8] "1195 | "9154 | °6834 | 4438
Q]} ‘tr51 | ‘orig | ‘6805 | °4416
°
I
2
3
4

aN
PwWPY HO O ONAN BOD HO

"1106 | °9082 | *6776 | 4304
"1057 | “9041 | °6745 | *4370
"0999 | °8994 | °6707 | *4342
"0935 | ‘8041 | °6665 | 4309
"0863 | °8883 | °6619 | °4274

ir

oe, Joint Life Annuity, at Top.
[169] Age of elder Life:— Last Survivor Annuity, at Side.

155

H™,
3 PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Inves.

94 95 96

0°6728 | 04129 | 0°1775
6735 | °4134 | £776
O74 | 4137 | °1777
"6744 | °4138 | °1778
6744 | "4139 | °1778

6743 | °4138 | +1778
"6739 | °4136 | °1777
"6732 | °4133 | °1776°
"6724 | “4129 | °1775
6716 | “4125 | 1773
"6712 | ‘4122 | °19772
"6709 | “4120 | °1777
°6708 | ‘4120 | °1771
"6709 | ‘4120 |) ‘1771
"6710 | *4a2r | ‘1777
"6710 | ‘412E | ‘1771
6709 | ‘4120 | ‘17971
"0707 | "4119 | "1771
6704 | “4118 | ‘1770
"6702 | “4117 | ‘1770
"6699 | “4115 | *1769
6698 | ‘4114 | °*1769
6696 | °4113 | °1769
"$113 | °1768
"6692 | ‘412 | ‘1768

"6690 | “4110 | ‘1768
*6687 | ‘4109 | ‘1767
°6683 | ‘4107 | ‘1766
6681 | “4105 | +1766
6678 | ‘4104 | °1765

"6676 | *4103 | °1765
"6674 | ‘4102 | °3765
6672 | “4101 | °1764
"6668 | “4099 | °1763
"6064 | *4097 | °1763

Lo)
Le |

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45 | °6658 | “4094 | °1762 | 45
"6651 | “4090 | ‘1760 ] 6
6644 | “4086 | 1759] 7
6638 | °4083 | 1758 | 8
"6631 | "4079 | ‘17561 9
50 | *6624 | *4076 | °1755 | 50
"6617 | °4072 | "1754 | 1
*66ag | °4068 | “17523 | 2
‘6599 | “4063 | ‘1750 | 3
"6589 | *4057 | "1748 | 4

se, | Joint Life Annuity, at Top.
[170] Age of elder Life:— Last Survivor Annuity, at Side.

156

H™,
8 PERCENT.
Values of Annutttes on Two Joint Lives, and on the Last Survivor
of Two Lnives.

55 24°6746 | 24°5239 | 24°3577 | 24:1792 | 239921 | 23: 7995 | “5
"6405 4890 | 3216] °1416| -9528| -7584
6077 4555 2870); -:1055 9150] ‘7188
5763 4233] -2538; -0709| -8788] -°6808
5463 | °3927| -2221 0380 {| °8442| °6445

5176| 3634) 1918] -0063; 8111 6096
"4905 "3357 1631 | 23-9763 | -7796| ‘5766
"4647 3094) °1359 9478 | ‘7496 5450
4403] -2845 1100| -9209| ‘7212 5151
4173) -2610] -0857| -8954| -6944| -4867

3956 | ‘2390; -0628 ‘8714| -6690; -4600
3752 2182; 0413; ‘8487; °6451 "4346
3561 1987) -0211 8275} *6226 4107
‘3380; -1803; -0021 8075 6014} 3881
3210] -1632 | 23-9843 ‘7887 | 5815 ‘3669

3052} =:1471 9677 7712| 5628] ‘3470
2906) 1324, +9524] -7551 5456 | 3285
‘2770| -1188} -9383| -7402| -5297 3115
2648 | °1065 9256} -7268)} 5154) :2960
2537 | 0954] -9141 7147} 5023) -2820

2437 0854} -9038] -7037 "4906 2693
2346} :0763 8944) -6938) -4799| 2577
2262 ‘0680 | -8859 6847 "4701 ‘2471
2184 0604; ‘8780; -6764| -4611 ‘2373
2113 0534 "8709 6689 4530} °2284

2049 0471 8646 | -6622 4457 | *2204
1992 0415 8589 6562 "4392 2133
1940 ‘0366 8538 6508 4334; -2069
1894 0321 8492 | -6460 4282 2012
1853 | -0282 8453 6418 4236; °1961

1816 0246 8416 6379 4194/ -1914
1781 0213 8383 6344 "4154 ‘1871
1746 70180} -8350 6309 4116 "1827
"1712 0148/ -8318 6275 ‘4079 1786
1677 0117 8288 | -6244 4044] +1746

"1641 0085 8258 6214 ‘4011 1708
"1609 | - :0057 8233 6188 3983 1677
1578| -0031 °8209 "6164 3959} +1649
"1550! °0007 8188 6145 "3939 "1626
"1526 | 23-9988 ‘8171 6129 3923; ‘1608

1507 ‘9971 8157 6117 ‘3910
1492 9960 8148 6108 3902

Ow w=
povu? coro

A
nv

ba
WW co~mr Ctr BO YH Hw CO LO CON OW

st

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“JI

5
6
7
8
9
fe)
I
2
5
4
5
6
7
8
9

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oo

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“fp, Joint Life Annuity, at Top.
(141) Age of elder Life:— Last Survivor Annuity, at Side.

— .

ae 0 rr re

157

H*,
3 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
16 17 18 19 20 a1

55 | 23°6047 | 23-4107 | 23-2199 | 23-0355 | 22-8572 | 22-6805 | 5
6 5616) -3658) +1734) 22-9874) -8077| -6299| 6
7 5203 | -3225| 1285 "9411 ‘7601 ‘O811| 7
8 4804) -2810| -0852 8965) 7144) 5342) 8
9 4424) -2412) -0439) +8537] -6704| °4893} 9

6o ‘4057| :2029) -0040 8126} ‘6281 4461) 60
1| .°3710| -1665 | 22-9661 7734] 5880} -4049|] 1
2 3378} -1317| -9299| -7360| -5496| -3657/ 2
3 3063; -0987} -8955|; -7004| -5131 3283 | 3
4 ‘2764! 0674} 8628] -6666/ -4783| -2929) 4

65 2481) -0376| -8318| 6345) 4455) -2592) 65
6 2214) 0094) 8023} 6040} 4142] -2273) 6
7 1960 | 22:9828 | °7745 ‘5751 ‘3846 1971) 4»
8 1721 0574| 7480) ‘5477| -3564| 1683; 8
9 1495 | -9335) -7229 5217} -3298| 1411] 9

70 1282] -9110) 6992} 4971) -3046| 1155} 70
r.| ‘1l085] -8901 6772) :4743/ -2811 0916) 1
2 0903] -8707;} 6568] -4530| -2593| -0693) 2
3 ‘0737 | -8530| 6381) -4336| -2394| 0490] 3
4 0586] -8369/ -6211 4159} +2213} 0305] 4

75 0449 | +8222} -6056 3998 | -2047| 0136] 75
6 0323] -8087) 5913] 3849] -1894/21:9981) 6
7{ -0208]} -7964) 5782! -3711| -1753| 9837! 7
8 0102] 7849) 5659; -3584/ -1622} :9704| 8
9 (0005 | =:7744/] 5547 3466) -1501 9581 | 9

80 | 229917] -7648| -5445| -3359) -1391 9469 | 80
I ‘9839 7563 5353 3263 "1292 "9368 | 1
2 9768) -7486| 5271 3176] +1203) 9278) 2
3 ‘9705 ‘T7417 5197 3099) -1123|} -9196) 3
4 9649] -7356 5132 "3029 1051 9124! 4

85 ‘9597 | -7299 5070} °2965 ‘0985 ‘9056 | 85
6 9549 | -7246| 5012} -2904/ -0923| 8993] 6
7 9500 ‘7192 4954 2843 0859 8928) 7
8 9453] -7140| 4897] °‘2782) -0797| ‘8865; 8
9 ‘9408; -7088; ‘4840| ‘2721 0734| 8801] 9

go | 9364] -7038) -4784| -2661| -0671| -8738] 90
1 | -9327] °6994] -4735) -:2607 0615; ‘8681; 1
2 9294] -6956; 4690] 2558) -0564| -8628] 2
3 "9267 | °6924| 4653) -2516| -0520; 8583] 3
4 ‘9246; -6899} 4623) -2483) -0483/ 8545) 4

95 9231 *6880| 4600; 2456; -0455) -8516]| 95
6 ‘9221 6868; 4585) -2439] -0436|) -8496| 6
[142] Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

158

H™,
8 PER-CENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

21-9431 | 21-7454 | 215436

5 22°5033 | 22°3224 | 22:1358
“4514

4016
3536
3076

2635
2214
1813
1432
1070

0728
0402
0094
21:9801
"9525

"9263
9020
8795
8588
8400

8229
8071
, 1926
‘7791
"7667

‘7553
‘7451
7360
7278
7204

‘7137
‘7072
7008
6944
6880

6816
6759
6707
6661
6624

6594
6574

So pwn nm

~

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6
7
8
9
fe)
I
2
3
4
5
6
7
8
9

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in bh DB Mm O

[143]

"2693
"2181
"1690
"1219

0768
0338
21:9929
"9539
‘9170

8819
8488
"8173
7875
7593

"7327
"7080
6850
"6641
6450

6276
6116
‘5968
5831
5705

5591
5488
5395
D312
5238
5170
5105
5040
4976
"4911

4848
‘4791
"4739
4694
"4657

4628
4608

Age of elder Life :—

0812
0287
21:9782
9300

8837
"8396
7976
‘T5577
‘7199

6840
6500
6179
5874
"5586

‘5314
5062
4827
4613
4419

424]
4078
3928
3788
3660

3544
"3439
"3345
3260
3186

3116
3050
"2984
2919

"2854 |

‘2790
2733
2681
2636
2599

2570
2551

8868
8328
“7809
7312

6836
6383
5951
5542
5153

4784
"4435
-4106
3793
3497

3218
2959
"2719
2499
2300

2118
1951
"1797
1654
1523

1403
"1296
1200
"1114
"1037

0966
‘0898
0831
‘0764
‘0698

0633
0575
0522
0477
0439

0410 |
0391

6874
6316
5781
5269

‘4778
‘4311
3867
3445
"3045

"2666
2306
1967
1645
"1341

1054
‘0788
0540
‘0315
0110

20°9923
‘9751
‘9592
"9445
9311

‘9188
9078
"8979
8890
8811

8738
8669
"8599
8531
"8463

8396
"8336
8282
8235
8197

8168
8148

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

4836
4260
3707
3179
2672
2190
"1731
"1296
0884

0493
0123
20°9773
9441
"9128

8832
"*8558
8303
‘8071
‘7859

7667
7489
7326
7175
7036

6909
6795
6694
6602
6521

6445
6374
6302
"6232
6161

6092
6030
5974
"5926
5886

5856
0835

wr
WO cOox~r Aw

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SO pwwn D

WO DG~3 An B&H He CO WO Cr QA

co co
© On Qn HH HHO

pow dS

‘oO
Cvwnr

159

H™,
8 PER-CENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.
28 29 30 31 32 33

55 21-3391 | 21-1324 | 20-9234 | 20-7122 | 20-4978 | 20-2806 | 55
2771 0682 8569 6433 4264 2064! 6
° 2176 0067 "7932 ‘5773 3580 1354] 7
8 "1605 | 20°9476 7321} ‘6140 2923 0674| 8
9 1058 8911 6737 "4536 2298 0024] 9
60 0535 ‘8371 6178 3958 "1699 | 19-9403 | 60
I 0038 “7857 5647 3410 1132 *8815| 1
2 | 20-9565 7369 5143 "2888 0592 8256] 2
3 | 9115] -6905| -4664| +2394] -0081| -7726] 3
Rt 8690 6466 "4211 1926 | 19-9597 7226 | 4
65 "8287 6051 "3782 "1484 9140 6753 | 65
6 "7905 5657 3377 "1065 8708 6306 | 6
7 "7545 5286 2993 0670 *8299 5884) 7
8 ‘7202 4933 2630 0295 7913 5484] 8
9 6879 4600 -2287 | 19°9941 ‘7548 5107) 9
70 6575 4286 1964 ‘9609 -7205 4754 | =o
I "6292 3995 "1664 9300 6887 44251 1
2 6030 3725 "1386 9013 6592 4120] 2
3 ‘5790 3478 "1132 8752 6322 3843 | 3
4 5572 3254 0901 "8515 -6078 3590 | 4
75 | -5374| -3050| -0690| -8298| -5855| -3361| 75
6 ‘5191 2861 "0497 ‘8099 5650 3149} 6
7 "5023 2688 0319 "7916 5461 "2955 | 7
8 4867 2528 0154 ‘7746 5287 2775 | 8
9 4724 2380 0002 “7590 5126 2610) 9
80 4594 -2246 | 19-9864 “7448 -4980 -2459 | 80
I 4476 -2125 9740 7321 4849 2324/ 1
2 4371 2017 ‘9628 "7206 4732 2203; 2
3 4277 "1920 9528 7104 -4626 "2095 | 3
4 4193 "1834 9439 7012 4532 1998| 4
85 | -4115] -1753| -9357| -6928| 4445] -1908] 85
6 4041 1677 9279 6847 4362 1823; 6
7 3967 "1601 -9200 6767 4280 1738 | 7
8 “3895 "1527 9124 6688 "4199 1655; 8
9 "3822 1451 9046 6609 4117 1571| 9
go 3750 1378 8971 6531 4037 1489 | go
I 3686 "1312 8903 6461 3966 "1415| 1
2 3629 "1252 8841 "6399 3902 "1349; 2
3 3579 "1200 8788 "6344 3846 1291| 3
4 3538 1158 | 8745 *6299 3800 1245] 4
95 3506 1125 8711 6265 3765 "1208 | 95
6 "3485 1103 "8688 6241 3740 1184] 6

[144] Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

160

H™,
3 PER-CENT.
Values of Annuities on Two Jornt Tives, and on the Last Survivor
of Two Lnves.

55 | 20-0603 | 19-8373 | 19: 6122 | 19-3855 | 19-1574 | 18- 9278
6 | 19-9832 ‘7571 5285 | -2983; 0664) -8326
‘9093 | -6802| :4484/ -2148|18°9792; -7416

‘8386 | 6066; °3718; 1349] -8959| -6546

W711 5364 | ‘2987; 0587;| 8164 5716

‘7067 4694 ‘2289 : 18-9861 ‘7407 4925
6456 4059 1629| -9172 6689 4176
5876 3457 1002; °8520| ‘6009; -3467
5327 ‘2886; -0409} -°-7903| 5366) -:2796
4808} °2348/18-9849; 7320) -4759| -2164

"4318 1839 | = 9321 6770 | °4187 1567
"3855 1358 | -8822 6252; +3648; -1005
3418 0905 "8351 ‘5763 | -3139 0475
3004| -0477 ‘7907. ‘5301 2659 | 17-9976
2614; 0073 ‘7488 | -4867 2208; -9506

2248 | 18:°9694 7095 4459 1784; -9065
1908; -9342 6730} -4081 "1392 8657
1594 9017 6393 3732 "1029 8281
1307 8720 | -6086 3413 ‘0699 "7938
1046 8450) 5807 °3124| -0400 7627

0809 ‘8206| 5554; -2862; 0128 ‘7345 |
0591 ‘7980 | 5320) -2620| 17-9878 ‘7086
0390 | <‘7773) 5106 "2399 "9648 | -6848
0205 ‘7581 4908|} :2194; °9437 6628
0034} ‘7405 "4726 2006] 9242 6427

18-9879 7244] 4560) -1834/ -9065 6244
"9740 7101 “4411 1681 8906 -6080
9615 6972} 4278; °1543/ -8765 5933
"9503 6856} 4159} :1420 8637 "5802
9404 6754} °4053/ 1311 8524 | 5685

"9311 "6659 | 3955 1210} -8420|] ‘5577
"9224! 6568) 3862 1113] °8321 5474
9136 6478 | -3768) °1017 8222 6373
9050 6389 | 3677 0923 "8125 5273
8963 "6299 | +3585 0828 8027 ‘5173

‘8879 6212 | 3494] -0735 ‘7932 5074
"8803 6133 | 3413; -0651 “7846 4986
"8734 ‘6063 | 3340} -0576| -7768 "4907
8675 6001 "3276 0510; ‘7701 "4838
"8627 5951 3224 | 0457 7645 “4781

"8589 5913; 3184) 0415/ °7603| :4737
8564} 5886) 3157) 0387) -7573| -4707

SO pepe
oO pwon

~

“J
© cCo~r An PON OE W/O Hr! O

c
.e)

5
6
7
8
9
)
t
2
3
4
5
6
7
8
9
fe)
I
2
3
4
5
6
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8
9

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-- {Joint Life Annuity, at Top.
[145] Age of elder Life: Last Survivor Annuity, at Side.

161

H™,
3 PER-CENT.
Values of Annutties on Two Joint Pires, and on the Last Survivor
of Two Ih
40 41 42 43 44 45
55 | 18°6966 | 18-4635 | 18-2286 | 17-9926 | 17-7566 | 17-5210 | 55
6 6971 3594 "1196 8782 6367 3952} 6
7 5019 2598 0152 ‘7688 | 5218 2747 | 7
8 4110 "1645 | 17-9154 6640 *4119 ‘1592/ 8
9 3242 0738 8201 5641 3070 70490} 9
60 2416 | 17-9872 ‘7295 4689 -2069 | 16:9439 | 60
1 1633 9053 6435 3787 1121 8442] 1
2 0891 8276 5622 *2932 0223 7497 | 2
3 0190 7543 4852 -2125 | 16°9375 6605 | 3
4 | 17-9530 "6852 "4128 "1364 8575 ‘763 | 4
65 8907 6201 "3445 0646 ‘7821 4969 | 65
6 8320 5587 -2802 | 16°9971 7110 42221 6
~| -7768| -5009| -2196| -9335| -6442| -3518| 4
8 7247 4465 "1625 8736 56812 2855] 8
9 6757 3952 "1089 8173 5220 2231) 9
70 6298 3474 0587 “7646 4666 "1649 | 70
1 6873 3030 0123 7158 4153 1109; 1
2 5481 °2621 | 16°9695 6709 3681 0611] 2
3 5125 °2249 9305 -6300 "3251 0159} 3
4 4801 "1912 "8952 5930 -2862 | 15°9749| 4
ye | -4508| -1606| -8633| -5595| -2509| -9378] 7°
6 "4238 "1325 8339 5286 -2185 9037| 6
7 3991 1068 8070 5004 "1889 *8724| 7
8 3764 0831 "7822 "A744 1616 8437) 8
9 | -3555| -0613| -7595| -4506| +1365] 8174] 9
80 3365 0416 "7389 4290 1138 7934] 80
I 3195 0239 "7204 -4096 0935 7720| 1
2 3043 0081 ‘7039 3924 0754 "7529| 2
3 ‘2907 | 16°9940 6891 3769 0591 -7358| 3
4 2786 9814 6760 3631 0447 7206! 4
85 | -2674| -9698| -6639| -3505| -0314| -7066] 85
6 2568 9588 6524 3384 0187 6933] 6
7 2463 9479 6410 3265 0062 6801] 7
8 2360 ‘9372 -6299 3148 | 15:9939 6672) 8
9 2257 9265 6188 3032 9816 6543] 9
go 2156 9161 -6080 2918 9697 6417 | go
I 2066 9068 5983 2817 ‘9591 *6305| 1
2 "1985 *8985 5896 2727 9496 -6205| 2
3 "1914 8912 5821 2648 9414 6118) 3
4 "1856 8853 5760 2585 9347 6047) 4
95 "1811 8807 6713 2536 9296 59931 95
6 "1780 8775 5681 2503 9261 5956| 6
Joint Life Annuity, at T
[146] Age of elder Life:— Last Survivor Anwuity, at Side.

162

H™,
83 PER-CENT.

Values of Annuttes on Two Joint Lives, and on the Last Survivor
of Two Lives.

iw
© COn~? An

1555 16: 9179 6830 “4503 2206 15- 9936
0290 | :7853; 5438] -3046| -0679) -8337

6580, 4104] -1645) 15-9211 6798
‘7919 | 5364) °2827) -0305 7804| °5322

6815 | :4203) -1607| 15-9024; +6458] -3909
5767; °3101 0448; -7806; -5179| -2564
4773 | °2056|15°9349| -6650| -3963| -1285
3834 | -1068; °8310} +5556] -2811 0073
2948; 0135) 7328) -4522|) -1723| 14-8926

"2113 | 15°9256| -6402| -3547) -0695| -7843
1325 | -8427| -6529/ -2627114:9726| -6821
0584; °7647 ‘4707; -1761 "8811 5856
15-9886; -6911 "3932; -0943| ‘7950| -4946
9229 -6219 3203; 0175 ‘7139 | -4090

8615 ‘5573 | +2521 | 14-9456 6380} -3289
8047 | °4973| -1890) -8790| -5677 2546
7523 "4.422 1309 ‘8177 | + =:5030| -1862
7046 °3920| -0780|) -‘7619| -4441 1239
, 6615 3465 0301 7114| 3908; 0675

6224 3053 | 14°9867| 6656] -3425| -0164
5864 "2675 9468) -6236 2981 | 13-9695
5535 2328; -9103} -5851 2574) °9265
5233 | +2010| -8767 "5497 ‘2201 ‘8870
4956 1718} ‘8460; 5172 1858 | 8508

4704) -1453|) 8180] -4878! -1548} -8179
4478) 1215 7930| 4614] +1269] -7885
4277| -1003; -7708| -4380| -1022| -7623
4097 0814{ -7508| -4170| -0801] -7389
"3937 0646} °7331 3983; -0604; -7181

3789 | 0490) 7168] +3811 (0422) °6989
3649) 0343) -7012| -3648] -0250| -6807
3510) 0197} ‘6859! +3486) 0080) -6627
°3374| ‘0054; °6709| -3329/13-9914]| -6452
3238 | 14-9911 6559 | +3170) -9747| 6277

3106; 9772) °6412/ -3017| -9586! -6106
2987 9647} -6281 2879 -9441 6954
2882 "9536 6164] 2756) -9312; 5819
2789 9438] -6062| -2649; -9200|} -5700
2715 9360} -5979| 2562) -9109| -5605

2658; 9300; 5916) -2496; -9039) -5532
2619 9258; 5872) 2450; -8991 5481

~r A
&Oo be O CO GO Qn RO YO =
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6
7
8
9
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I
2
3
4
75
6
7
8
9
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2
3
4

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CO
ak pwnd OO Cr~r Qn

oe, Joint Life Annuity, at Top.
O47] Age of elder Life :— Last Survivor Annuity, at Side.

163

H™,
8 PERCENT.

Values of Annutties on Two Joint Inves, and on the Last Survivor
of Two Inves.

—

52 53 55 56 «57

5 | 15-9429 | 15-7304 | 15-6225 8: 8:01

8 7695| -+5493| +3336 949 368, °3
4 | -6022| -3743| -1507 14-7191 7
8 | 4410} -2055 | 14-9742 5265 | 14-3113] 8 |
9 | -2861| -0432/ -8042 3405| -1171] 9!
60 | -1878|14-8875| -6409 ‘1615 | 18-9299 | 6o
1 |14°9964| -7390| -4849 139901} -7505|
2| °8619| -5975| +3363 | 8262] -5787) 2
3| -7342| -4631] -1949 6700} -4147| 3
4} 6135] -3359} 0609] - ‘5216| 2588} 4
6s | 4993} -2154/ 13-9339] - 3808 | 1106 | 65
6 | -3913] -1015| -8137| - 2471 | 129698} 6
+ | -2895|13-9939| -7001| - -1205| 8363] 4
8} -1934| -8923] -59297| - 0007| -7097| 8
9} 1028] -7966| -4914| - 128874] 5900] 9
vo} -0181| -7069| -3965| - ‘7810| °4775| 70
1 |13°9395| -6237| +3084 -6821| -3728| 1
2| -8671| +5470] -2271 5908) -2760| 2
3| 8011] -4771| -1530 5074] -1876| 3
4| °7414| -4138| -0858 ‘4318| -1073| 4
ye | -6873| -3664| -0249 3631} 0344 | 76
6| -6375| -3036|12-9689| - -2999 | 11-9672] 6
»| -5920| -2553| -9176 2419] -9055} 7
8| -6502] -2109| -8704 1885| -8488| 8
9} -3118| +1702) -8271 1396} -7966| 9g
80 | -4770} +1332] -7878 0951 | -7492]| 80
1 | 44571 -1000| 7525 0551| -7067| 1
2| +4180} -0706| -7212| - 0196| -6688| 2
3| -3932| -0442] -6932 119878] -6349| 3
4| °3711] -0207| -6682 ‘9594| 6047] 4
8s | -3508]12-9991| +6452 ‘9333 | 5768) 8°
6| -3815| -9786| -6233 9084| -5502| 6
»| -31294| -9584] -6018 -8839| -5240| 7
8| -2939| -9386] -5809 ‘8600| -4985! 8
9 | +2753| -9189} -5598| - 8360| -4728} 9
90} °2573| -8998| -5395| - 8129| -4481] 90
1| -2411| +8827) -5213| - ‘7999| -4260| 1
2| -2268| -8674| -5051| - ‘7738 | -4063| 2
3) -2143| -8542/ -4910| - ‘7678| -3891| 3
4| 2042] -8435| -4796| - ‘7448| -3753] 4
95 | +1965} -8353| -4710 ‘7350 | +3648] 95
6| -1912| +8297] -4650 ‘7982 | -3576| 6

1. Joint Life Annuity, at Top.
[148] Age of elder Life:— Last Survivor Annuity, at Side.

[149]

58

8°3274
"1790
"0233

13°8998

7043
5167
3368
1649
0012

12°8454
6972
5565
4230
"2965

1776
0667
11-9642
8705
7853

‘7079
6364
5708
-5105
"4550

“4044
3591
3187
2826
2503

2205
1921
1641
1368
1093

‘0828
0591
0380
"0197
0048

10-9936
"9858

59

8°1323
7°9920
"8445
“6898

13°4852
‘2891
1010
12-9210
7494

5858
"4299
‘2818
"1409
‘0074

11-8817
‘7645
6559
5566
4662

"3840
"3081
"2384
"1741
1150

0612
0128
10°9698
"9312
8967

8649
8345
8046
7753
"7459

“7174
6919
6693
6495
6336

6215
6132

Age of elder Life:—

164

H™,
8 PER-CENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

6o

7°9319

*7986
"6593
*5129
3590

13-0686
12°8721
"6839
6041

3325
1688
0129
11-8647
7238

“5911
4671
"3523
‘2471
"1513

0641
10°9835
9093
8410
7780

‘7207
"6691
6232
5821
5453

‘5113
4789
"4468
"4155
3839

3534
‘3260
3016
2803
2632

2501
"2412

61

7°7242
"5996
"4683
"3301
"1847

0328

12-6508
"4543
2662

0865
11-9148
‘7511
5951
"4468

"8068
"1759
0545
10-9431
8417

7493
6637
5850
5123
4454

"3843
"3293
2804
‘2365
"1973

1611
1264
0922
‘0587
0248

9-9920
"9626
9364
9134
8949

8809
"8712

62

75127

"3958
"2724
°1423
"OO51
6°8616
7125

12:2326
0362

11-8483
6684
"4967
3328
"1768

‘0293
10°8912
"7630
6453
-5380

"4401
3495
2659
1888
1176

0526
9-994]
9420
"8952
8534

"8148
7778
"7413
7054
"6692

6341
6024
5743
5495
5296

5144
5040

Joint Life Annuity, at Top.
Last Sarvivor Annuity, at Side.

63

7°2972

°1879
0723
6°9501
"8210

6857
°5449
"3989

11-8145

6182
4302
‘2503
‘0784
10°9145

7593
6138
4786
3543
2409

1374
0414
9:9528
"8710
7954

“7264
6641
6087
5589
5143

4732
"4338
3948
3565
"3178

2802
2463
‘2161
1895
1680

"1517
1404

tn
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fe,
NO pup nd

~T
WO @~71 An H © WB ew O WO CO~ Qn

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CO co
8 WO wo AN P&H HH O

ve)
An B&O HD HH

165

H™,
3 PERCENT,
Values of Annuities on Two Joint Lives, and on the Last Surviwwor
of Two Lives.
64 6, | 8 | 6
55.| 7°9787 6'°3969 | 6°1587| 5°9130] 55
6 | 69767 "3155 | °0837| °8441] 6
7 "8686 "2288 "0038 "7700| 7
8 "7542 "1366) 5°9185 6921] 8
9 "6330 "0383 *8276| °*6082] 9
60) °5059 5°9348| °7314) ‘5194 60
I 3734 "8262 6306 "4260] 1
2 2357 "7129 "5252 "3283/1 2
3} 0933 5952] 4155] °2265] 3
4 "4735 | “3019 | ‘“I210| 4
65 | 11:3968 "3470} °1836| ‘orog| 65
6 "2005 "21471 °0597| 4°8954| 6
7 0124] = -7827 | 10-5612 4'9292| ‘77341 7
8 | 10°8324] -5945| -3646 | 10-1428 6441] 8
9 6604] -4143/ °1760| 9-9457] 9-7234 9
70 4974) -2432) 9:9968] -7581| -5272| 9:3040) 70
I 3443 | +0824) -8280; ‘5811; -3418} -°1101] 1
2 2019] 9:9326) 6705) -4157] -:1683} 8:9283] 2
3 0708] -7945| 5252; +2629; -0078] -7597) 3
4] 99511 6682 | :3921 "1228; 8-8603| :6047/) 4
75 8417) -5527| °2703) 8:9943| -7250) :4623| 75
6 7402| +4454) +1569] -8746] -5987| 3290] 6
7 6464| -3462} 0520) -7637) -4815| °2052| 7
8 0597 | -2544| 8:9548| -6608| -3726; 0900; 8
9 "4796 | -1694) ‘8647; -5654| -2715| 7:°9828] 9
80 4062} -0916} °7823; -4779! -1786| °8842} 80
I 3401 0214{ -7077) +3988] -0946| -7949] 1
2 2812} 89588] 6412} -3280) -0194| -7150; 2
3 2282 9025 | 5812) -2643| 79516) °6428) 3
4 "1808| -8520; 5275] +2072} -8908| 5779) 4
85 1369} -8053} 4778] -1541 8342} 5175} 85
6 0949 ‘7605 | °4300) -1031|- -7797| °4593) 6
7 0534) °7162| -°3828| -0526| -7257; 4013] 7
8 0126| -6728| 3364] -0030| -6724| ‘3442; 8
9 | 89713] -6287| -2893| 7:9526; -6184| -2860/] 9
go 9312} -5859 2436} -9037| 5659] :2293] 90
I 8949 | +5472 2023; +8596; 5185 W781 | 1
2 8626; +5127 1655 | -8202; -4762) +1324] 2
3 8341 4822) 1329) -7854| +4389; 0922) 3
4 8110] -4575 1065| °7572| +4087 0596) 4
95 7935 | -43887| -0864| -7357) °3857| -0348] 9°
6 7813| -4256| 0724) -7207 3697 | -O175| 6
[150] Age of elder Life:— Joint Life Annuity, at Top.

Last Sorvivor Annuity, at Side.

166

H*,
3 PERCENT

Values of Annusties on Two Joint Lives, and on the Last Survivor
of Two Lnves.

S41 Od
"3530
"2914
"2255
"E547
"0797
"0005

49175

"8308
"7497

"6464
°5470
4416
"3293
"2081

"0796

8-4778
2930
"1225 | 7:8998

7-9654 | -7353
8178 | +6807
6802 | -4361 |
‘5517 | -3009 |
4318 | -1744 |

3211 | :0575
2204 | 69510
"1302 | °8554
0485 | ‘7689
6°9750 | -6908 | - "1704 | 5-9349

9064 | -6179 | -: 0890 | -8494
8399 | ‘5471 | - 0097 | ‘7659
7734 | 4762] - 5°9299 | -6817
‘7075 | -4056 | - 8500 | -5972
6399 | -3330 | - ‘7672 | 5094

5737 | :2615 | 5- “6852 | 4222
5134 | °1963 | - 6098 | 3419
4595 | -1377 5417 | = =:2692
4117 | -0855] - 4805 | -2037
3730 | 0431 | = -73 4303 | -1498

3435 | -0107 | - 3916 | °1082
3230 | 59881 | - 3644 | -0789

nr
in

i
F nw on8 oor ai |
OD puww nS Oro

6
|
8
9
70°
I
2
3
4

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06
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. { Joint Life Annuity, at Top.
[151] Age of elder Life: — ( Last Sarvivor Annuity, at Side.

167

H™.
8 PER-CENT

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

3°3947
"3715
"3465
"3193
"2895
"2576
"2236
"1876
"1497
"1103
0687
"0242
2°9763
"9240
"8655
"Borg
"7322
6597
"5858
"5131
"4424
"3690
2938
‘2160
"1339

jr
© co~r A pw eH OO DAT AG

~

°
I
2
3
4

75
6
7
8
9

5-2851
1486
0241

49113

‘8051
7002
5929
4836
3676

2504
1408
0398
39477
"8709

"8109
‘7682

oe Joint Life Annuity, at Top.
(152) Age of elder Life:— Last Survivor Annuity, at Side.

CO
pwpn d © won ake b&w = O

\O
Own

168
H™

3 PER-CENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

82 83

mn |

3°0129 2°6876 | 2°5429
2°9943 | ° 6728

79742 | ° "6566

"9523 | ° "6390

9282 | ° 6196

"9024 | ° "5986
"8748 | ° "5762
8455 | "5524
"8147 | ° "5274
"7826 | ° "5013

"7488 | ° "4739
"7126 | ° "4446
6735) ° "4126
"6305 | ° "3774
"5820 | ° "3374
°5285 | ° 2928
"4704 | * "24.39
"409r | ° "1920
"3462 | ° "1386
2843 | ° "0858

"2241 | ° 0345
"1612 | ° "9807
70966 | ° "9250
"0204 | I' "8668

1'9580 | ° "8045
8835) ° "739%
"8089 | ° 6731

"6102

4°7083 *5503

‘5875

4737 4°1986
"3611 0793
2454 39564
"1268 *8300
3°9998 | - 6936

8702 | : 5536
TATT | 4205
6340 | : "2958
5289 | ° 1793
4403 | - 0802 | 2°9197

3705 |: 70011 | °8368
3202 | - 2°9434 | °7762

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,. 9 Joint Life Annuity, at Top.
[153] Age of elder Life: Last Survivor Annuity, at Side.

169

HM,
3 PER-CENT,
Values.of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

1°6568 | 1°4235
"6512 | “4192
"O450 | “4145
"6384 | “4004
"6308 | °4036

6224 | °3971
"6134 | °3901
"6036 | °3826
"5934 | °3740
*§828 | °3664

"5719 | °3579
"5604 | °3490
"5480 | °3394
"5342 | °3289
"5179 | °3164

"4989 | °3017
"4771 | *2847
4532 | 2657
"4279 | °2455
"4029 | °2256

"3791 | °2067
3536 | "1864
"3271 | °1653
2986 | °1427
"2668 | ‘1172
"2319 | °0888
"1952 | ‘0587
"1594 | °0292
"1247 | ‘ooo1
"0943 | 9°9747
0676 | °9526
70418 | 9315
70132 | ‘9084
0'9824 | °8840
9356 | *8461
eee e o8
2°4363 19
2708 | 2:0712
"1141 | 1-9005
19791 | °7525

8701 | -6322
‘7897 | 5429

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Joint Life Annuity, at Top.

[154] Age of elder Life:— ) 7 ost Survivor Annuity, at Side.

170

H™,
3 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Laves.

}

55 | 0°6577 | .o'4051 | 071746 | 55
61 *6564 | *4044 | 1743 | 6
7| 6550 | ‘4037 | ‘1740 ) 7
8 | *6535 | ‘4029 | 1738] 8

| 9 | 6517 | °4020 | ‘1734 ] 9

6o | *6497] ‘4010, ‘1730 | 60
1' *6476| °3998: 1726] 1
2' 6452 | °3986| ‘1721; 2
3; °6427 | °3972 | “1716 ) 3
4 | “64or | °3958 | ‘r7Ir b 4

65 | °6374 | °3944 | °1706 | 65
6; *6346] °*3929] ‘1700 | 6
7 | 6316 | “3913 | -1694| 7
8: 6285 | °3897 | °1688| 8
9 | °6247 | °3877 | °*1681 | 9

70 | *6202 | °3854 1 ‘1672 | 70
1 | ‘6149 | °3826| :1662/ 1
2 | °6087 | °3793; "1650| 2
3 | *6018 | °3756 | °1636| 3
4/ °5949] °3718 | ‘1621 | 4

75 | °§886 | °3684 | +1608 | 75
6) °5817 | °3646] ‘1594 ] 6
7 | °5746 | 3608) "1579 | 7
8 | °5671 | °3567 |} °1564| 8
9 | 5585} °3521 | 1546] 9

Bo | °5485 | °3467 | +1525 | 80
I) °5374 | °3405 | ‘I50r |] 1
2| "5264 | °3344 | ‘1478 | 2
3 | °5149 | °3279 | °1452 | 3
4} °5048 | 3221 | ‘1429 ] 4

85 "40061 *3172 "1409 | 85
6} 4881 | °3127 | °1392 | 6
7: “4795 | °3°77 | ‘1370 | 7
8 | 4720 | °3038 | 1357] 8
9 | “4591 | °2970 | °1332) 9

9° | °4378 | *2850] °1285 | go
I “4117 2702 | °1226|
2 °3798 | °2524 | *1157 2
3; ‘3311 | ‘2240 ] ‘1042 | 3
4 | —— | ‘1849 | *0879 | 4

9 09073 | ——— "0647 5
3 7674 | 05288 %

[155] Age of elder Life:— Teint Life or Amity, ot de.

—
————

TWO LIVES,

SS
= -—————

H™,

THREE AND A HALF PER-CENT.

172

H*™,
33 PERCENT.
Values of Annuities on Two Jctnt Lives, and on the Last Survivor ~
of Two Lives. Equal Ages.
I
z Gy .z 24,—Gz,2 ad | Azz
10 19°3289 24-5789 55 ' 85546 14°5424
I "2029 4655 6 | "2385 "1805
2 0346 3432 7 | 79231 13-8135
3 18°8322 "2134 8 | 6085 4417
4 "6054 0770 9 | 2944 "0656
15 "3635 23-9351 60 | 6:9834 12-6860
6 "1158 "7894 I 6767 "3037
7 17°8722 6410 2 "3754 11-9196
8 "6423 4913 3 “0805 5341
9 "360 3422 4 5°7931 1483
20 2554 "1938 65 515 10-7611
I "0876 "(1440 6 | +2342 3726
2 16°9259 22°8915 y] 4°9595 9-9827
3 "7606 ‘T7344 8 6863 ‘5909
4 "5843 5715 9 401i 1965
25 "3949 4023 70 "1378 8-8028
6 "1960 "2270 I 3°8695 4123
7 | 15°9893 0461 2 | ‘611K 0287
8 7798 21-8606 3 "3676 76556
9 "5689 6703 4 "1437 2961
30 "3561 ‘4751 7 2°93 69499
I "1419 2751 2 188 6098
2 14°9231 0693 7 "5543 2773
3 "6988 20-8578 8 "3730 5-9518
4 "468 5 6399 9 "1949 6321
35 "2329 4161] 80 | 0225 3219
6 13°9930 "1860 1 | 1°8612 0258
7 "7502 19-9504 2 "7174 4°7486
8 "5047 7085 3 "5886 4876
9 "2549 "4603 4 *4806 2438
40 12°9996 "2056 8 380 ‘0079
I "7363 189431 7 tel 3°7671
2 "4629 6729 7 2107 -5109
3 "1805 3947 8 1202 -2366
4 11°8g21 "1093 9 0036 2°9242
45 "5979 17-8165 go 0°8625 5809
6 "3019 ‘5177 I "7193 2269
7 "0062 2132 2 5762 1-8608
8 10°7102 16-9026 3 4173 “4709
9 "4122 5858 4 2654 ‘0798
5° "1123 "2625 95 1314 0°6944
I 9°8082 15-9322 6 0326 "3224
2 "4986 ‘5944
3 "1854 ‘2498
4 88707 | 14-8991

173

H™,
33 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survtvor
of Two Inves.
10 II 12 13 15

19°'2650 | 19°51 783 | 19°'0722 | 18°9516 | 18°8209

10 10
1 | 24:5231 "1178 | °0136| °*8949| ‘7661| 1
2 "4645 | 24:4053 18°9323 | °8157| ‘6890| 2
3 4045 | °3434 | 24:2794 "7177 | °5932| 3
4 3435} +2805) 2144] 24-1463 “4833 | 4

15 *2823| +2174; :1492| -0789 | 24-0072 15
6 2216} -1549} -0847! -0120 | 23-9379 | 23-8635 6
7 ‘1620| -0934) -0212| 23-9463] 8699) -7930) 7
8 ‘1039 | -0336 | 23-9593 | -8823| -8036| :7242| 8
9 0481 | 23-9759; -8997| -8206|) -7397] ‘6580; 9

20 | 23:°9943| -9204| -8422| -7611| 6780} -‘5941/| 20
I 9418] -8662| -°7862| -7030| 6178) 5317) 1
2 "8901 8129; -7310} °6458| 5585] -4702; 2
3 8387 | °7598) -6761 ‘5890; 4995) -4089) 3
4 7874) 7067} 6211 *5319| 4404) 3476) 4

25 ‘7357 | °6536; ‘5661 ‘4748 | -3811] :2862| 25
6 6842; -6003) ‘5111 4178) °3218) :2246) 6
7 "6331 5474| 4562) 3611) :2629; 1633) 7
8 5823| -4953; 4022) -3049) -2047/ :1028| 8
9 5321 4435 | °3489) 2495) ‘1471 0430; 9

30 4826{ 3924] 2959} 1949] -0902 | 22:9838) 30
I ‘4337| +3420; °:2438] 1406] 0342} 9255) 1
2 3852 | °2921 1922} -0871| 229784) -‘8678| 2
3 8372} :2426| 1410) -0342| -9234; 8105) 3
4 2895 | -1935) -0903 | 22°9815| +8688] ‘7537) 4

35 2424| -:1451; -0403/ 9296) -8149| 6976] 35
6 1957 | :0970|22°9907| 8782] -7614| 6421; 6
7 1498; -0499/ -9418| -‘8277| -:7089 5875 | 7
8 1044; 0032; ‘8936; -7776| 6570) 5335) 8
9 ‘0597 | 22:9571 "8461 7284) -6059; 4804) 9

40 0155! 9118} -7992| 6798) -5554| °4279! 40
1 | 229718] ‘8669; -°7529| -6317) °5055| 3759) x
2 9285| 8226] ‘7071| 5843) -4561] °° 3247) 2
3 *8859| -7787| 6619] ‘5374| -4074| 2739] 3
4 8439) -7356| 6173) 4913) :3594) -2240) 4

45 8025} -6933| 5737) -4458| 3123] +1749) 45
6 7621 6518} 5309| °4016; ‘2661 1269} 6
7 ‘7227| -6114| °4892] -3583) 2212] -0800; 7
8 6843 | 5720] 4485] 3160; -1772| 0343; 8
9 6467| -5336' -4087| -2749| +1343] 21:9895) 9

50 "6101 ‘4961 3701 2346 | 0924] -9459] so
I 5744] 4596! -3323| +1955); -0516) 9033] 1
2 5397) 4239) °2955| +1572) -O117| 8616) 2
3 5058 | 3892| 2597} -1200| 21:9728] 8210) 3
4| °4731 ‘3557 | -2250| -0839| -9352] -7816| 4

we, _ § Joint Life Annuity, at Top.
[188] Age of elder Life:— } y ast Survivor Annuity, at Side.

174
Hm

32 PER-CENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor

of Two Lives.
16 17 18 19 20 21
10 | 18°6849 | 18°5485 | 18°4168 | 18°2949 | 18°1842 | 18°:0779
| *6319° °4974| °3674) 2474} °1384] 0338
2| °5508 °4243| °2964/ °1783| 0713 | 17'9685
3 | °4634 +3331] °2073} °0913| 17°9863| °8856
4| °3559 °2279] 1044] 17°9906; -8878| °7892
15 | °2384) “1129 | 17°9919 8804) °7798) 6834
6 | £7°9927 "8742 7651 "6668 "5720
7 | 23°7165 | 7560} °6494| °5534| "4015
8 | +6452 | 23-5674 5379| °4443| 3547
9 ‘5766 :4963 | 23-4180 3445) °2573
|
20 5104, 4278) +3471 | 23-2692 "1703
I 4458 3609] °‘2779| -1976; 23-1201
2 3820. +2949) +2095) °1270| -0472 | 22-9689
3 3185: -2290) 1414] :0566) 22:9744|) -8939
4 2548| 1630} ‘0730/ 22:°9859| -9014| ‘8185
25 ‘T910| 0967; 0044; -9148| -8280| °7427
6 ‘1272 -0306 | 22°9357| °8437} -7546|) 6668
7 0635 | 229646; °8673) ‘7728; -6813] 5912
8 0006| °8992| 7996; -7028; -6088| ‘5163
9 | 22°9384| °8346| °7326) °6335|) -5372| 4421
30 ‘8770 | -7708| ‘6664; 5649) -4663| 3689
I 8163; -‘7078| 6010] 4972] 3962) +2965
2 ‘7563 | 6453] °5363| -4300| +3268, -:2247.
3 6969| °5837| 4722) -3638|) -2581/ -1538
4 6377 | +5224) 4086) -2978| -1901|) 0833
35 ‘5795| 4618] 3459] +2328] -1228) 0139
6 5218| -4019| +2836) -1685| 0563 | 21:9451
7 ‘4650| -3429] +2225) °1051/ 21-9909} -8776
8 ‘4089; °2846) °1621| 0426] -9262| -8109
9 3536 | :2272| :1025/21-9810| 8626) 7452
40 -2991| -1706| -0438| -9202| -7998| -6804
I "2451 1145 | 21-9856 | -8600| -7377| °6163
2 ‘1917|- -0590} -9281| -8005| -6764| ‘5530
3 1389; ‘0041 8712| 7418] 6157] 4906
4 ‘0870 | 21:9502| 8152] -6839| 5561) -°4291
45 0359| 8971) ‘7602; °6270| ‘4974; -3688
6 |21:9859| -8451| 7063) 5713} 4401] -3097
7 9372| 7944} -6538|; 5170! -:3841 "2522
8 8894| +7449} 6024) 4640; -3295| -1960
9 8429} 6963) ‘5522; -4121 2761) 1411
50 ‘7974 | 6491 5030 | °3614] +2240; -0876
I "7529| °6029| 4551 3118) 1730) 0353
2 7095| 5576} 4082) -2634] 1231 | 20-9841
3 6671 | 5135) +3624; -2161 0746) :9342
4 6260| ‘4706; °3180] -1602| -0274/ -8859
. Joint Life Annuity, at Top.
[189] Age of elder Life:— Last Survivor Annuity, at Side.

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23 24 25 2.6 27 |
|
IO | £7°9725 | 17°8627 | (7°7444 | 17°6168 | 1774812 | 17°3385 10
I | °9300| °8219) °7054| °5792] “4454] "3045 «1
2 "8666 "7603 "6457 "5214 "3893 "2504 2
3 | °7857| °6813) *5688) 4466] 3165] “17941 3
4| 6914) °5892|) °4787| 3587] *2309| ‘0960 4
15 | °5878| °4879| °3796) ‘2617| +1362] 0037 15
6| °4793] °3816| +2757] +1602] ‘0369| 16:9068! 6
7 | ‘3704) 2751) 1715] '0585| 16°9375| <“B097! 7
8 | *2660| 1729] °0717} 16°9610| *8426| 7172! 8
9 "1708 | ‘0800 16°9811 8729| °7569| °6340 9
20 ‘0861 | 16°9977| ‘gor! "7952 6815 | ‘5610, 20
I 0056} ‘OI94| °8252 "7217 "6105 "$923 | I
2 8420} 7502] °6492) *5404| ‘4247! 2
3 | 22-8142 6712| °5727| °4665| *3533/ 3
4| °7364| 22-6542 4883 | °3847| -2743! 4
25 6581 | 5734 | 22-4882 2940 1864.| 25
6 5798} -4925| -4047 | 22-3161 "ogi2| 6
7 6017| °4119| -3213] +2299 | 22-1380 7
8 4243) +3320] :2387| +1444] -0497/21:9549| 8
9 "3477 | +2528) -1569| -0598/ 21-9621) -8643] 9
30 2719! 1744) 0758} 21-9760| -8754| -7746| 30
I 1972] -0971 | 21:9958| -8932| -7897| -6858) 1
2 1231} 0205; -9165| -8110] -7045| -5976/ 2
3 ‘0497 | 219447) -8380| -7297| -6203| -5103/| 3.
4 |21:9769| -8694|) ‘7601 76490] 5367) -4236/ 4
35 9051 ‘7951| °6833| -5694| +4549 3380 | 35
6 8341) -7217| -6072| +4906] -3795| -2534) 6
7 7642) -6495| -5325! +4132] -2922| -1700 | 7
8 6953) 5782} -4588| +3369] -2129| -0879;| 8
9 6276| 5082) -3863;) -2617/ -1351/ -0069| 9
40 5607 | 4392; +3149} -1877| -0584| 20-9274] 40
I 4946) -3710) 2444] +1147 | 20-9826| -8488 , 1
2 4294) 3036) +1748) 0427] -9080} -7712, 2
3 "3650 | 2372) -1061| 20-9716] -8344) -6949| 3
4 3017} 1719) 0387) -9018| -7621| - -6200| 4
45 ‘2396 | = 1079 | 209726 | -8335| -6912| 5465] 45
6 1788; 0453) 9080] -7667| -6221 4748) 6
7 "1196 | 20-9844 | -8452| 7016) +5548] 4051] 7
8 0619) -9249] -7839| -6384] +4892] -3372| 8
9 0055 | °8669/ -7241| 6766) -4254) -2709| 9
50 | 20°9504| -8104) 6658) -5165/ -3632| -2066 50
I 8968; -7552} -6090| +4578] -3025/ -1438} “1
2 8443| -7013| 5535] -4006| -2434| -0827| 2
3 7932} -6489/ -4996/ -3450/ +1859) -0232| 3
4 ‘7436| -5980) 4473] -2910) -1302/19-9656| 4
[190] Age of elder Life:— Joint Life Annuity, at Top.

175

H™,
34 PER-CENT.

Values of Annuities on Tuco Joint Lives, and on the Last Survivor
of Two Lives.

Last Survivor Annuity, at Side.

176

H®,
33 PER-CENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lnves.

32 33

|

Io | £7°1918 | 17°0414 , 16°8869 16°7287 | 16° 5049 16°3950| 10
I "1591 "0103 8574 «=9'7007| = °5383 "3699; 1
2 "1069 | 16°9596! °8086! °6536| ‘4929] °3262| 2
3 | °0381| "8929 °7435) 5907) *4319| *2669| 3
4 | 169567 | 8137; 6666) "5155} °3590) 1901) 4

15 "8667 29 “5811 | 323 "29797 "II7ZI| 15
6 "7722 6338 "4912 «= °3448 "1925 °0340' 6
7| °676| *5416, ‘aorg 2573] *1075 1} 15°9512| 7
8) °5874 4538 | | +3160! +1743] +0267 “8729! 8
9| 5065) 3752 ‘2398, ‘10041 15°9553| “8036 9

20| 4360] "30701 “3739, 0369] 8940] *7448| 20
1 "3697| °2433| °1125,15°9778| °8373] °6903) 1
2 "3046 "1806 "0524: "9200 7818 "6373; 2
3| '2357| ‘1143'15°9887! 8589} °7232| “5811) 3
4 "1594 0406! ‘9177 | "7906 76576} °5182| 4

25 | 0744] 15°9584| °8382| 7139} °5838) °4472| 25
6 |15'9820| ‘8690, °7517| °*6303| 5032] 3695) 6
7 | *8830| °7730| °6587; “5404) °4163) 2857] 7
8 6728 "5617 | "4405 "3254) ‘1980| 8
g | 21°7670 "$609, °3489] ‘2311 "1070| 9

30 “6741 | 21:5743 | "24.73 "1328 | ‘o120| 30
I ‘5822 | 4792 | 21:3768 | "0307 | 14°9133| 1
2| -4910) -3847| -2790 21-1740 Bo91| 2
3 -4005| 2909} °1819, :0735 | 20°9654 3
4 3106) -1978 0854 | 20°9735| 8619] 20°7507| 4

35 2218| °1057;20-9900| -8746| 7594] 6446) 35
6 1340} 0146) -8954| -7769| 6578) 5392] 6
7 0476 | 209250) -8024| ‘6800| ‘5576| 4353) 7
§ | 20:9623| -8365| °7106| +5847] 4586] -3326| 8
9 8784} -7494| -6201 ‘4909| ‘3611 2313 | 9

40 ‘7957| °6635| 5310) -3983) :2651] 1315] 40 |.
I ‘7141 5788| 4430; -3069; 1701 0328; 1
2 6337 | °4953| :3562| -2167| -0764|19-9354) 2
3 5543; 4129) -2707| -1278/19°9840] 8393) 3
4 ‘4766| °3322| 1868} -0407| ‘8934| 7451) 4

45 4004} ‘2531 1046 | 19-9553 | 8046] -6527| 45
6 -8960| +1759} -0245| -8720| -7180| 5627) 6
7 2537] °1008| 19-9466) 7911 6339 | 4751) 7
8 1833| -0278| ‘8707| -7123; 5520} 3899) 8
9 1146| 19-9566] -7969} 6356) -4723} -3070| 9

50 0479] 8874] ‘7251 5611] °3949| :2265) 50
1 119-9829] 8200] 6552} ‘4886; 3195; 1481) 1
2 9196} 7544) °5871 4180| :2462] 0719) 2
3 8580] 6907] 5211 3494| -1750)18:9979| 3
4 ‘7984| -6290| 4572) -2832} 1062) -9264/] 4

Joint Life Annuity, at Top.

(191) Age of elder Life:— Last Survivor Annuity, at Side,

177
H™,

33 PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Inves.

a

) Ge)
I
2

3
4

3

we

ew

&

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> FH Ye OO OO DO AN MOH KH OC YW COm™*QAN BO HY =O OO CO™ AW

in
PwWPYHO OO ON AN

[192]

16° 162186
"1949
"1528
"0955
"0266

15°9498
"8691
7884
[124
"64.55
"5887
"5307
"4860
"4323
*3720

"3038
‘22.90
"1483
"0638
149760

"8844
"7802
"6885
"5818

20°5300
4208
3130 !
"2065 |
1014

19°9975
*8949
7937
6938
5957

"4996
“4058 |
3147
"2259
"1397

0558
18-9742
"8949
"8179
"7436

“16'0360 0360 | 15° 8497
"8267

"0136
15°9731
"9177
"8508

"7762
"6976
"6193
"5454
"4808

"4263
"3704
*3281
"2769
‘2191

"1537
"0818
"0042

14°9229

"8384

"7 501
"6584
"5613
"4582
"3487

20°3031
1914
‘0808
19-9717

8639
"7572
6518
5479
4458 |

3458

2481 |

"1531
‘0607
18-9708

"8834
‘7984
7158
"6356
5982

Age of elder Life :—

‘7877
"7341
"6693

"5967
"5203
"4442
"3727
“3101

"2578
"2102
"1641
“1153
"0602

14°9975
"9285
8538
7757
"6945
*6007
“5211
4279
°3286
"2229

"I109

20-0703
19-9557
8423

7304
6196
‘5099
4018
2956

"1914
0897
18-9907
"8944
8006

7095
6209
5347
4512
“3704

15°0544
"6346
"5974

“5454
"4826

"4121
"3379
"2640
"1946
"1343
“0840
"0385
14°9948
"9483
"8957

"8357
"7696
"6980
"6229
"5449
"4035
"3788
"2889
"1933
"OOT5

13°9834
"8695 |

19°8316
7140

5978
"4827
"3688
2562
1456

‘0371
18-9312
‘8280
‘T7276
6298

5348
"4423
3525 |
"2653 |
1812,

——

15° 4500
"4376
“4019
"3518
"2908
"2224
"1503
"0786
"0113

14°9531
"9050
"8615
"8200
"7759
7257
"6683
"6052
"5304
"4045
"3897
*3116
"2304
"1442
0523

13°9543
"8503
"7404
"6253

19-5867

4660
3464
2280
"1110
18-9959

8829
1725
6650
5603
4583

"3592
2627
"1689
‘0780
17-9901

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

39

15°2518

°2347
"2004
"1520
"0929
"0265
14°9566
"*8870
"8219
"7957
"7196
6782
6387
"5969
"5492

"4945
"4340
"3684
"2904
2278

"1531
0752
3°9927
"9046
"Bro4

"7 104
"6048
"4939
°3775

19-3353
‘2111
‘0881

18-9665
8467

7290
6140
‘+5019
3928
2864

"1829
"0822
17-9842
8892
‘1974

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178

Hx,
34 PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

>
YO con Avr
&>
wh
qi
|

aN

— O&O YH & O

19-0768
18-9490
8226
6980

"5755
“4556
3388
2249
1139

0059
17:9007
7983
6991
6031

aN

i
hOwNYH O WO Cr AN

[193]

18-8106
6791
"5495

4220
‘2972
1754
0566
17:9407

8279
‘7180
6109
‘5071
4068

Age of elder Life :—

13°9710

"9238
"8714
"8144
‘7544
"6922
"6273
"5597
"4877
"4108
"3284
*200
1475
"0494
12°9465
"8374

"7215
"5970

18-:5365
‘4017

2689
1389
0120
17-8880
‘7670

6491
5342
"4223
3136
"2085

"0993

"0723
"0322
13°9825
"9260
"8663
"8071
7523
"7959
° "6692
"6374
8: "6077
" "5793
7594} = °5399
"7146 | °4975
6647} °4501
6104} °3984
"5535 "3443
“4943 | °2881
4325 | °2295
“3683 "1685
"2998 | "1035
2266! °0339
1480 | 12°9592
0642] ‘8794
12°9753 "7946
"8817 | = °7054
"7832 "O114
6787 "5116
"5676 |) *4053
4482 | °2909
*3190| °1669
0336

18-2547
‘1166 | 17-9657
17°9812 | °8248
8488 | °6869
‘7195 "5521
5932 “4202
“4700 | °2915
3498 "1659
"2327 0433
1190 | 16-9242
0088 -8089

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

13°8586

8481
"8224
"7842
"7361
6816
"6239
"5667
5138
"4693

"43.44
4042
"3763
"3468
"3125

"2723
229 5
"1784
"1270

"0737

"0182
12°9604
"8988
°8328
“7618

"6859
"6053
"5204
"4399
"3358

°2343
"1249
"0062
11°8782
°7422

17-6700
5264
"3858
2483

1139
16°9826
"8544
"7297
6088

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179

H™,
$i PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

Io

bag

PWHHO O DOYIOAn BWHN HO © CONAN BO HHO © BIA BW YHHO

"5922 73325 686| ° 3255
"5678| -3094| - “5002
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"4849 | .2297 . . "4362
"4322 | ‘1790; ° "3908
"3765 | "I25r| * ; "3426
3213) “O7IO) ° " "2949
"2703 | ‘'0227/ ° 51: "2512
2276] 12°9818| ° . "2151

"1943 | ‘9502| ‘7 "447: "1880
"1659| ‘9233 . . "1656
"1397| °8988| °65: . "1457
"II20| °8728/ ° . "1245
"0797 | 8424) ° " "0995

"0417 "8067 , ° "0695
12°9992 *7664 ° ° 0357
"9527, °7223] ° "9985
"9040 "6762 ° . 9597
8535] °6285) - "9196

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"5582 | °3492| °1347, ° "6358

4862 | -2811| 0702" "6285
*4096{ °2085| ‘oors los "5074
"3289 | ‘1320/1 11°9291| ° "5029
"2439 | 0513) 8527) ° "4348
"1534|11°9654} °7712! ° "3621
-0568| 87351 6841! - 2841
11°9523| °7740| “5895| ° "1992
*B388| 6656; °4863| ° "1062
7162) °5485| °3745| ° "0050
"§857| °4235| ‘2550; ° 10°8966
4470] 2905 "1278 | 10° 7807
"1511 | 10°9942 8: 6
17°3684 ° Been 6317
2220 | 17-0609 "5. 3988
‘0786 | 16-9116 | 16°7473 "2591

16-9384 | -7654{ -5950| 16: —_
8012 6223 | °4457 , 16°1006
6672 4822 | °2995 "1192 | 15-9416 | 15-7666
5367 "3457 "1569 . "7862 6045
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«- , _§ Joint Life Annuity, at Top.
[194] Age of elder Life:— Last Survivor Annuity, at Side.

180

H™.
$i PER-CENT,
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

(Oo | 11°9607 | 11°6657 | 11°3657 | r1°O611 | 10°7528 | 10°4410

95608 | °6626/ °3634) ‘0595| 7518] 4408
"9399 | 6468) °3488|) ‘o460} °7393| 4293
"Qr2i 6204} °3238| ‘0224! ‘7171 "4084
8760} °5860| ‘2909] 10°9910} °*6873| *380r

"8342 | °5459| °2526) ‘9544] °6523| 3468
-7896 "5031 "2115 "OIS! "6147 *3109
"7455 | °4607| ‘1709| °8761| 5774] °2752
"JO5T| «4220 ] =°1337| 8406) °5434] °2428
"6722 "3906 "1038 "8120 "5163 "2168

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*6480| +3676] ‘o821 "7916| °4969} °1987
"6282 | 3492] "0648) °7754| “4819) °1847
6109 | = °3331] 0500} "7617| 4693) “1731
"5927 | °3162) °0344| °7474| "4561| ‘1610
"5799 | °2959| °O155} 72909; °4399| “1460
"5445 | °2712 |) 10°9925 | °7083| “4198|) 1274
"5140| °2432| *9662| °*6837/ °*3968] 1058
4815} ‘2121 | 9370 0564 | "3712| ‘o819
4471] °1798) 9067) = °6280 3447} 0571
4217) 1465) 8755} 5989; °3175| °0318
"3750| ‘“X121| °8433| °5688; ‘2894] 0056
"3379 | °0767| “8102! °5379| *2607| 9°9790
2965] 0388) °7749| "5049| °2300| "9505
‘2529 | 10°9980| °7368| 4695) 1969] ‘9197
"2058; 9539] °6955| ‘4310, ‘1612) °8864
1552] 9065] °*6512| °3896| 1226] °8507
"10131 8559] Gogo} 3455] ‘o815] “8124
"0443 8026} 5540] ‘2990| ‘0382 "7721
10°9842 "7462 "5014 | °2498] 9'9925 "7297
"9199 | “6860; 4451) 1972] "9435| °6843

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i ©] °7753| °5592] °3178} 0784) °8327| “58tt
2 ‘6921 "4719 "2443 °0094| ‘7681 "5209
3 | “Gorg¢; 3862) *1637| 9°9337| °6972| °4545
4 °5039 "2941 0767 "8519 "6204 "3826
45 | °3993] 1951} 9°9833| °7638| °5376| 3049
6 "2891 "0907 "8846 6708 "500 "2228
7| 1740] 999816} °7814) °5734| °3584| °1366
8 "0534 "8671 "6730| ‘4710 "2619 "0461
9 | 99263} 7463) 5585] °3627| 1598] 8:9500
50 "7923 *6188 "4374 "2481 "0516 "8480
r} "6501| °4833] °3086| 1259! 89359} *7389
2 "3386| 1707} 89949] °8118} °6215
3 | 15-4255 °0246| °8558! 6796} °4963
4 ‘2607 | 15-0779 "7OQI | “5401 } °3639

oe, Joint Life Annuity, at Top.
[195] Age of elder Life:— } 7 ost Survivor Annuity, at Side.

181

HM”,
81 PERCENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lnves.
58 59 6o 61 62 63

IO |10°'5263 | 9°8083 | 9°489t | 9°1694 | 8°8502 | 8°5324 | Io
1 | +1266! ‘8093 | ‘4905 | °1714 | °8527 | °5353] 1
| a] 1162| "7998 | -4819 | 1637 | “8459 | “5293 | 2
3 | °0965 | °7814 | °4647 | 1477 | °8309 | “5155 | 3
4} *0697 °7561 ; “4409 | "1251 | “Bo98 |) “4956 | 4
15 | ‘0380 | 7258 | ‘4122 | ‘og8o | °7841 | °4714 | 15
6 | °0037 | °6932 | ‘3811 | °0685 | °7561 | ‘4449 / 6
7 | 9'9697 | -6607 | “3502 | ‘0390 | -7282 | 4183 | 7
8 | -9387 ; °6313 ; °3221 | °0124 | °7027 | °3943) 8
9] ‘914t | *6079 | *3000 | 89914 | °6829 | °3754| 9
20} ‘8969 |} °5918 | ‘2849 | °9772 | °6696 | °3630 | 20
1] *8840 | °5797 | ‘2737 | ‘9669 | “6601 | °3542 | 1
2 | °8733 | °§700 | ‘2648 | °9588 | °6528 | +3476] 2
3 | *8623 | "5599 | °2557 | "9595 | °6453 | °3499 | 3
4] *8485 | “5472 | ‘2440 |) °9398 | °6355 | 3320] 4
25 | ‘B31r | °5§312 | ‘2291 | ‘9261 | ‘6229 | °3203 | 25
6 | ‘8112 | °5125 | ‘2119 | ‘gr00 | *6080 | °3066 | 6
7 | °7888 | °4917 | ‘1924 | ‘8920 | “5912 | ‘2910 | 7
8 | -7656| -4701 | “1724 | °8733 | 5740 | *2750] 8
9 | °7420 | 4482 | ‘1519 | “8544 | “5564 | ‘2588 |) 9
30 | °7178 | °4256 | ‘1311 | °8351 | °5385 | 2423 | 30
1 | +6930 | 4027 | ‘1099 | “8155 ) °5205 | °2257 | 1
2 | *6665 | °3783 | °0873 | °7947 | °5013 | ‘2080 | 2
3 | 6381 | "3519 | "0629 | °7722 | “4806 | 1890] 3
4 | ‘6072 | *3233 | °0365 |) ‘7478 | “4581 | °1683{ 4
35 | +5739 | 2924 | ‘oo80| “7215 | °4339 | ‘1460 | 35
6 | +5385 | ‘2596 | 89776 | °6934 | ‘4080 | ‘1222 |] 6
7 | ‘gor "2251 | "9457 | “OO40 | °3810 | *og74 | 7
8 | -4618 | °1887 | ‘gr21 "6331 "3526 | ‘o714 | 8
9 | °4196| ‘1497 | °8762 | ‘6oor | °3222 | °0435 | 9
40 | "3742 | '1076 | °8373 | °5643 | 2893 | °0134 | 40
I} °3239 | ‘o61t | *7943 | °5246 | 2528 | 779798} 1
2| ‘2678 | ‘oogo | *7460/] “4799 | ‘2116 |] ‘O419 | 2
3 | °2059 | 89513 , “6924 | “4302 | "1656 | “Boo4 | 3
4 | 1387 | ‘8887 | °6341 | “3701 | “1155 ) “8531 | 4
45 | *0660 | *8208 | ‘5709 | °3173 | ‘0610 | °8026 | 45
6 | 89891 | "7490 | *5039 | "2551 | “0031 | “7491 | 6
7| °9084 | °6736| °4337 | °1897 | 779425 | “6929 | 7
8 | -8234 | "5941 | 3596) ‘1208 | °8785 | *6337 | 8
9} °7333 | °5097 | ‘2808 | °0475 | ‘8105 | °5706| 9
§0 | *6376 | “4201 | ‘1970 | 7°9693 | 7378 | °5033 | 50
I} 5349 | °3237 | “1068 | “8851 | “6594 | “4304 | 1
2| ‘4242 | ‘2196 | ‘0092 | °7938{ °5741 | “3511 | 2
3 | "3059 | “1082 | 779045 | “6957 | °4824 | "2055 | 3
4 | ‘1806 | 79899 | *7932 | *5912 | °3840) °1742) 4

[196] Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

182

H™.
$i PER-CENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Inves,

a]
=

7°5885 | 72739
"5925 | ‘2782
5887
"5780
"5619

"5417
"5104
"4968
"4764
4606

"4504
"4436
4.388
"4341
"4275

"4185
"4077
"3954
"3828
"3701

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"3444
3308
*3162
"3002

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«¢, (Joint Life Annuity, at Top.
[197] Age of elder Life:— ({ Last Survivor Annuity , at Side,

188

Hm,
$4 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lnves.

6°3206 | G'oogg | 5°6966 | 5°3998 | 51185 | 4°8528
"3255 | “o100 | *7017 | “4050 | *1237 | *8580
"3241 | ‘OOOI | °7012 | °4049 | “1240 | °8586
"3171 | *0028 | *6957 | ‘4001 | ‘1198 | °8549
"3054 | 5°9921 | *6860 | °3913 | ‘1118 | °8478
"2903 | °9782 | °6732 | ‘3794 | ‘1009 | ‘8378
"2730 | °9622 | °6583 | °3657 | ‘0882 | °*8260
"2550 | °9458 | °6430 | °3514 | *0749 | *8136
"2395 | °9308 | °6290 | °3383 | ‘0627 | ‘8022
"2272 | "9193 | °6183 | °3283 | °0533 | 7934
"2197 | ‘9123 |- 6118 | °*3223 | °0478 | °7883
"2349 | *9080 | 6079 | °3188 | °0446 | °7854
"2120 | "9055 | °6057 1 °3169 | ‘0429 | °7839
"2092 | ‘9031 | °6037 | °3151 | ‘o4r5 | °7827
"2051 "BQ94 | “6004 | °3123 | °0390] *7806

"1990 | *8939 | 5955 | °3°79 | ‘035! | °7771
"1915 | °8871 | *5893 | °3023 | *0300 | °7725
"1827 | -8791 | “5821 | °2957 | °0240] °7670
"1738 | *8709 | °5746 | °2889 | °or78 | °7614
"1648 | 8628 | °5672 | ‘2822 ] ‘ors7 | °7558

"1559 | 8547 | °5599 | °2755 | "9056 | °7503
"1471 | °8467 | °5527 | *2690 | 4°9098 | °7451
"1378 | °8384 | “5451 | °2622 | °9936 | °7395
"1278 | °8293 | °5369 | °2548 | ‘9869 | °7335
"1169 | "8194 | °5279 | °2467 | °9796 | °7269
"1050 | °8087 5183 "2380 "9717 "7197
0926 | °7974 | °*5081 | 2288 | °9634 | °7122
0797 | °7858 | 4976 | ‘2194 | "9549 | °7046
"0664 | °7739 | °4870 | ‘2098 | ‘9464 | °*6969
"0524 | °7613 | °4756| °1997 | °9373 | °6888

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49 | °0372 | '7477 | °4635 | °1888 | ‘9277 | 6802 | 40
I | *0200} °7322] ‘4497 | °1765 | ‘9166 | ‘6703 | 1
2} ‘ooo | °7143 | ‘4335 | °1620 | ‘9036 | ‘6586 | 2
3| 59775 | °6938 | “4151 | °1453 | °8885 | "6451 | 3
4] °9525 | °6712 | °3946| *1268 | °8718 | ‘6300 | 4

45 | "9249 | °6462 | "3719 | ‘1063 | °8532 | °6132 | 45
6 | *8957| ‘6197 | °3479 | °0845 | °8335 | °5954| 6
7 | °8653 | "5931 | °3229 | ‘0619 | “8131 | °5769 |] 7
8 | °8332 | °5631 | ‘2967 | ‘0382 /] *7918 | °5576} 8
9 | ‘7990 |} °5321 | °2687 | ‘0130 | °7690 | °5371 | 9

50 | °7624 | *4989 | °2387 | 4°9860 | °7446 | °5152 | 50
1] ‘7224 | ‘4627 | *2060| °9564| °7179/] ‘4911 ] I
2| ‘6782 |} ‘4225 | ‘1696}| °9234 | °6881 | ‘4641 | 2
3 | *6299 | °3786 | ‘1297 | °8873 | °6553 | °4345 | 3
4} °5778 | °3311 | *0865 | ‘848t | °*6198 | *4023 | 4

oe Joint Life Annuity, at Top.
[198] Age of elder Life:— Last Survivor Annuity, at Side.

184

Hm,
8} PERCENT.

Values of Annutties on Two Joint Inves, and on the Last Survivor
of Two Lnves.

4°0946 | 3°8524 | 3°6173 | 3°3941

10 | 4°5929 | 4°3406 10
r | *5980 | °3457 | "0995 ; ‘8572 | *6219 | °3986] 1

2| *5988 | °3467 | ‘1008 | "8586 | *6234 | “goo2 | 2

3 | °5958 | °3441 | ‘0986: “8568 | 6220) “3990 | 3

4| °5893 | 3384 | °0935 , “8523 | "6180 | 3955 | 4
15 | *5802 | °3301 | “0859! 8455 | "6118 | °3900 | 15
6 | °5693 | °3201 | °0768 , ‘8371 | “6042 | °3830 | 6

7 | °5578 | °3094 | ‘0669; “8281 | "5959 | °3754 | 7

| 8| 5472 | 2995 | °0577 | °8195 | “5880 | -3681 | 8
| 9] °5399 | ‘2919 | *0506 | “8129 | “5819 | °3624 | 9
| 20 | +5342 | -2875 | -o465 | “8092 | *5784| 3592 | 20
| ¥] "5345 | ‘2850 | ‘o4g3 | “8071 | °5766) °3576| 1
| 2 | 5303 | 2840] +0434 | °8064| 5759] “3570 2
| 3 | 75294 | '2832 | *o428 | “8059 | °5750 | °3568 | 3
4| °5275 | ‘2816 | cogrs | “8048 | °5747 | “3560} 4
25 | °5244 | ‘2789 | ‘0390 | °8026 | °5728 | "3544 | 25

_ G| °5203 | "2751 | *0357 | °7997 | *5701 | 3520} 6
| 7 | °5453 | 2706 | *0316 | °7960) °5668 | “3490 } 7
8 | ‘5ror | ‘2659 ] °0274 | °7922 | °5633 | "3459| 8
' g| ‘5051 | °2614 | °0232 | °7884 | °5600 |] °3429] 9
| 30 | ‘5001 | ‘2569 | ‘org2 | 7848) °5567 1 °3399 | 30
1] "4954 | °2526 O14 | "7814 | °5530 | °3372] 1
2] ‘4904 | ‘2481 | corr3 | °7778) °5504| "3344 | 2
«3 | *4850 | °2433 | "0070 | °7739 | °5479 | °3312 | 3
| 4 | 4799 | °2379 | ‘ooar 7695 | “5431 | °3278 | 4
35 | 4726 | *2321 | 3°9969 | °7648 | 5389 | “3240 | 35
| 6} 74658 ‘2260 | ‘gor4 | °7599 | °5344| °3200| 6
7| 4589 | °2198 | ‘9859 | “755° | “5300 |) ‘5161 | 7

| 8] ‘4521 "2137 "9804 | ‘7501 "5257 "3123 8
9| "4449 | °2073 | °9748 | ‘7451 | °5213 | “3084 ) 9
| 40 | "4372 | *2005 | "9687 | °7398 | °5166 | -3043 | 40
r | 4284 | 1927 | ‘9618 | ‘7337 |} ‘5112 | ‘2995 | 1

2| ‘4179 | °1833 | °9535 | ‘7263 | °5047 | °2937 | 2
3} °4057 | ‘1724 | ‘9437 | °7175 | °4969 | ‘2868 | 3
4 | "3921 | “1601 | °9327 | °7077 | “4881 | ‘2790 } 4
45 i 3769 | °1464 | °9204 | *6966 | °4782 | ‘2701 | 45
6 | *3609 | “1319 | °9073 | °6849 | °4677 | ‘2607 | 6
7, °3442 | “1170 ]) ‘8939 | °6729 | °4569 | ‘2510 |] 7

8 | °3269 | *1014 | ‘8799 | *6604 | *4458 | ‘24rxr | 8

g | °3084 | °0848 | °8651 | °6471 | *4340 | *2306 | 9
50 | °2887 | °0671 "8493 | °6330 | °4214 | ‘2194 | 50
1 | ‘2670 | ‘0477 |) °8318 | °6175 | °4076| ‘2071 I

2| ‘2427 | *0258| °8122 | ‘6000 | °*3920, ‘1932 | 2

| 3 | ‘2159 | “007 | *7906 | *5805 | °3746 | "1777 | 3
| 4 "1868 | 3°9755 | °7670 | °5594 | “3557 | “160B |) 4

e, Joint Life Annuity at Top.
[199] Age of elder Life:— Last Survivor Annuity, at Side.

185

H™.
$i PER-CENT.

Values of Annuities on Two Joint [ives, and on the Last Survivor
of Two Lives.

82 83 84 85 86

om
O O37 An BW DH =A O
o=s

Go

"1886 | 2°9980 | 2°8261 | 2°6652 | 2°5048 | 2°3350

"1928 | 370022 | °8301 | ‘6691 | °5085] °
"1945 | °0039 | °8319 | °6709 | “5104
"1936 | °0033 | ‘8315 | °6708 | “5104
"1906 | *0006 | °8292 | °6688 | *5088
"1856 | 299961 | °8251 | *6652 | *5056
"1792 | °9903 | °8198 | ‘6604 |] *5013
"1722 | “9838 | “8139 | “6549 | *4963
"1654 | "9776 | °8081 | °6495 | “4913
"1602 | 9727 | °8035 | °6453 | °4874
"1572 | ‘9700 | ‘8010 | ‘6429 } 4852
"1557 | °9686 | °7997 | °6418 | “4841
"1553 | 9082 | °7994 | “6415 | °4840
"1552 | 9682 | °7994 | 6416 | “4841
"1545 | °9677 | *7990| °6413 | °4839

"1531 | "9664 | °7979 | “O4o4 | °4831
"1510 | °9645 | *7963 | °6389 | *4818
"1482 | °9621 | °7941 | °6369] °4800
"1454 | 9596 | °7918 | °6349 | °4782
"1427 | “9571 | ‘7896 | °6329 | 4764

"1401 | "9547 | °7874| °6310 | °4747
"1376 | °9526 | °7855 | ‘6292 | *4732
"1351 | 9503 | °7835 | °6274 | °4716
"1323 | '9478 | °7812 | °6254 | °4699
"1292 | "9450 | ‘7787 | °6232 | °4679

"1258 | "9420 | °7760 | ‘6207 | °4657
"1223 | °9388 | °*7731 | °6181 4634
"1187 | "9356 | °7703 | °6156] °4611
"1153 | °9326 | °7676| °6133 | “4591
"I1IQ | °9296 1} °7650| ‘6109 | ‘4570

we
|
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40 | °1083 | °9264| ‘7622 | °6085 | ‘4550 40
I} ‘lor | ‘9227 | °7590 | °6057 | 4526 I
2{| *o9gg0 | ‘9182 | °*7550 | *6022 | °4495 2
3] *0928 | ‘9127 | *7501 | °5979 | °4457 3
4 | 0858 | 9064 | °7445 | “5929 | “4413 4

45. | °0778 | “8993 | °7381 | °5872 | °4363 45
6} "0694 | ‘8917 | °7313 | °5811 | *4309 6
7 | ‘0608 | ‘8840 | °7244 |] .5749 | "4254 7
8 | ‘os1g |- 8761 | °7173 | °5686 | -4198 8
9 | °0426 | °8678 | ‘7099 | *5620 |] ‘4140 9

50 | °0327 | ‘8590 | ‘7021 | °§551 | *4080 50
1] °0217 | "8492 | *6934 | °5475 | ‘4013 i
2 | "0093 | °8382 | °6836| °5387 | °3936 2
3 | 2°9954 | °8258 | °6725 | °5289 | “3850 3
4 | °9803 | °8123 | “6604 } “5182 | °3755 4

oe, Joint Life Annuity, at Top.
[200] Age of elder Life:— Last Survivor Annuity, at Side.

186

H™.
83 PER-CENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lnves.

1°9449

79478 | ° "4628 | ‘2108
9495 1° "$042 , ‘2118
"9500 | ° $648 "2124
"9494 | ° "4046 9 *2.123

"9476 * "$637 «9°28 |
9449 | ° "$621, °2106 |
"9416 | * "$600 ‘2091
"9380 | ° "$576 °2072 |
"935T 5 * "$555 °2056

"9335: ° “4543 °2047
9327 | ° "4537 2042
"9326 | ° "4537 , °2041
"9328 © °6 "4539 | °2045
"9329 | * “$540 ; °2044
"9325, ° "$538 *2043
‘9317 * “4534: "2040
9306 | 6952] -4527 , 2035
"9294, * "4519 | °2029
"9283 . "4512 | ‘2024

9272 | ‘692: "$505 | *2019

bo to
iO C~T AWN #W Hm CO WO DW~?1} An FW H=™ OO OO CON? AN Sw WB =O

Go
Oo

"9263 | “6917 | “4500! ‘2015
"9254 | - “4494 | “2017
"9243 . . "2006
"9231 | ° "2001

>

"92t7 | ° ; "1995
"9202 | ° ’ "1988
"9188 | ° . "1981
"9T75 | * "4445 | “1974
"9163 | °68: "1969
“OI5T | ° "44 "1964
“9139 | * ; "1959
‘Ot2r | "0005 | * "1953
"9099 | * ; “1943
"9072 | ° "1932
"9041 | . "1918
"9007 | ° "4! "1902
"8973 | ° : °1886
"8939 | ° | °1871
"8903 . . 1855
"8867 | ° : "1839
"8827 | ° "423. "1822
8781 | °653. ’ "1802
"8729 ‘ . ‘1778
"8675 e ° "1753

‘ -., _§ Joint Life Annuity, at Top.
[201] Age of elder Life: Last Survivor Annuity, at Side.

as
& i)
O CAF An BWwWYH HO OO ONTO BW HNO © COM AN HW BHO

mv Hm O
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187

H™.,
$i PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

—
.@)
ro
~~]
fe, )
nN
rr)

0°6684 | 0°4107
6692 | ‘4111 | °1768
"6697 | “4114 | °1769
"6700 | ‘4116 | °1769
"6701 | 4116 | °1770

bond

6699 | “4115
6695 | ‘4113 | *1769
"6689 | “4110 | °1768
"6681 | °4106 | *1766
"6673 | ‘4102 | *1764

Le |
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bug
O D7 An Bw HHO

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OO CO~I CAG HPWH WHO WO CON? At BW H HH OC WO CDHr~ Qtr WD =O

6668 | ‘4099 | ‘1763
"6666 | ‘4097 | “1763
6665 | *4097 | °1763
"$097 | °1763
"4098 | °1763
"6666 | ‘4098 | °1763
‘6666 | ‘4097 | ‘1763
"6663 | °4096 | ‘1762
"6661 | *4095 | 1762
6659 | “4094 | °1761
"6656 | ‘4092 | ‘1761
6655 | ‘4092 | ‘1761
6653 | ‘4091 | ‘1760
"6651 | ‘4090 | ‘1760
"6649 | °4089 | ‘1760

"6647 | °4087 | °1759
"6644 | °4086 | °1759
"6640 | °4084 | °1758
"6638 | *4083 | °1757
6635 | *4081 | °1757

wv

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AX
AD
AD
“I A

40 | *6633 | ‘4080 1756 | 40
t | °6631 | "4079 | ‘1756]| 1

2| ‘6629 | ‘4078 | ‘1756 | 2
3| *6625 | "4076; "31755 | 3

4 | 6620 | "4074 | 1754) 4
145 | “6615 | ‘4071 | "1753 | 45
6 | "6608 | ‘4067 | ‘1752 | 6

| 71 *6602 | ‘4064 | ‘1750 | 7
8 | *6595 | ‘4060| 1749) 8

9 | °6588 | 4057 | "1748 | 9
50 | °6582 | *4053 | ‘1746 | 50
1} *6575 | 4050] "1745 ) 1

2) °6567 | *4045 | "1744 | 2

3} *6557 | ‘4040 | °1742] 3

| 4 | "6546 | '4035 | ‘1740 ] 4

Joint Life Annuity, at Top.

[202] Age of elder Life:— Last Survivor Annuity, at Side.

188

H™,
34 PER-CENT.
Values of Annutties on Two Joint [ives, and on the Last Survivor
of Two Lives.
ra | 12 | 13 14 15

22-4413 | 22-3232 22-1914 | 22-0489 | 21-8987 | 21-7434] ¢¢

8 a i08 2919 -1591| -0152] -8634| -7065| “6
7 | 3812! +2617. 1279] 21-9827} -8294| -6708] 7
8 | -3527| -2327° -0978| -9514] -7966] -6364 8
9 | 3256] 2049-0601) 9214" “7651 6035] 9 |

60 | -2995| +1784 -0417]} -9998 -7350| -5718| 60 |
1| -2747| -1530' -0154| 8653 = -7063| 5415} 1
2} 2512] -1290'21-9905| -8394 -6789| -5127/ 2
3| 2288] -1062; -9669| -R146 -6529/ -4852] 3
4| °2078| 0847, -9447) -7913 -6283| -4592] 4 |

6s {| 1877! -0643! -9235| -7691 -6049| -4344 6s
6| -1688' -0451 -9036| -7482 -5827] -4110| 6
7 | +1511) -0271, -8849| -7285 5619] -3888] -
8} -1343| -0100, -8672| -7099 -5421| -3678] 8
9 | -1184] 21-9939: -8505) -6923 5934} -3479| 9

7o| 1036] 9790: -8351| -6760 -5061| 3293] +o
1] -0899| -9651! -82907| -6609 -4900| -3120| ‘;
2| -0772| -9524! -8076] -6470 -4751| -2960| 3
3| -0657| -9408| -7956| -6343' -4615| -2815| 3
4| -0553| -9304; -7848| -6229 4493] -2683| 4 |
ve | -0458| 9209; -7750| -6126 -4381| -2562 6 |
6] 0371] -9123! -7662) 6031. 4280] -2452) "6
7 | 0291) -9043; -7580/ -5945 4186] -2380| 7
8| 0217} 8971; -7505| 5866. -4101| -2258| 8
9) 0150) -8905| -7438; -5795 4024! = -2173| 9 |

80 | -0088} -8845' -7377| -5730. -3954| -2097| 80 :
r | 0033} -8791| -7322| 5673-3892] -2028] 1 |
2 |21-9983| +8744] -7274| -5622 -3836| -1967| 2
3| -9940| -8701; -7231| +5576, -3787| -1913] 3 |
4| -9900 8663; 7192} 5535 3742) 1864) 4

85 | -9865| -8629| -7158| +5498 -3702| -1819 gs |
6| -9830} -8596| -7124] +5463 -3663| -1776| 6
7| -9797| -8565| -7093| -5429: -3626| -1734| 7
8 | -9764] +8534] -7063| +5398 -3591| +1695] 8
9g} 9729 8503 ‘7033| 5367 -3557| -1656| 9

go] 9694] -8472; -7004| -5337 3525] 1618 | go |
1} -9662| -8445| -6978| ‘5311. 3497] -1587|] “1 |
2| --9632| -8419} -6956,; 5289 3474 ts 2
3| 9604| -8396} 6935, 5269. 3453) +1537] 3
41 9581] 8376] -6918 5254' -3437| +1520! 4

95 | 9561] -8360| -6904 5241 -3425| 1507! gs
6 6

9548 "8349 6895 "0234 | "3417 "1498

oo Joint Life Annuity, at Top.
[173] Age of elder Life:— Last Survivor Annuity, at Side.

\

189

H™,
$i PER-CENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

|
8
|e] |

55 | 21-5860 | 91-4290, 21-4290 | 21:2747 | 21° 1256 20-9815 | 20-8389

wn

6) 5474] -3887| -2329 2
7 3100; 3497] -1923 4
8 4740; 3120} 1532 8
9 4394; -2759' “1155 9
160, ‘4062! -2411: -0794 60
1 °3743| -2078, -0446 I
2 3440, -1759| 0116 3
3 °8150| -1456| 20-9798 3
4 | 2876 1168 "9499 4
1651 -2614| -0893| -9211 65
| 6 2366] -0632! -8938 . 6
7; 12132] 0384: -8680 7
! 8 1908| -0149; -8433 8
ig, 1696} 20-9924} 8197 9
yo, 1498} -9713' -7976 70
1 | +1313} -9517' -7769 I
2 1142 ‘9335 T5T7 2
3 | 0985} -9168| -7401 3
4: 0843} 9016; -7240 4
a6 | -0713| -8877| -7093 ”5
6' 0594] +8749! -6957 6
7 | 0483] 8630] -6831 7
| g | -0382| -8521| -6715 8
Ig | 0290| +8420} -6608 9
180 02061 -8329! -6510 80
| r 0131) 8247! -6429 I
' 2 -0064| +8174! -6344 2
i 3. 0004! -8109! -6273 3
| 4 | 20-9950 ‘8049 | -6209 4
/ 8s -9900! -7995| +6151 8s
| 6. 9852] -7942| -6094 6
' 71 +9805] -7890| -6038 4
| S| ‘9760| -7840| -5983 8
' 9 | 9716] 7789; +5927 9
'go! -9673| -7740| -5872 go
“1 | 9636] -7697! -5893
2| -9605| -7660| -5781
3| -95771 -7628) -5743
4 | -9557| -7603| -5713

\o

I
2
3
4
5
6

95 | 9542| -7585! 5691
6 ‘9532 7573 | ‘5677

oo Joint Life Annuity, at Top.
[174] Age of elder Life:— Last Survivor Annuity, at Side.

190
H™,
$i PER-CENT.

Values of Annutnes on Two Joint Lives, and on the Last Survwor
of Two Lives.
1

24 s | 26 |

55 | 20-6955 | 20°5486 | 20-3965 | 20-2388 | 20-0763 | 19-9098 | 55
6\ -6489| -5009| -3475| 1883} -0242| -8560| 6
| 7 {| +6039] -4548| 3002} -1395|19-9740| 8041} 7
| 8 | -5605| -4103| 2545) -0926| -9954. -7540| 8
9g | -5187| 3676) -2107| -0474) -8790; -7060| 9
| 60 | 4786 3265] -1686| -0042; 8343; -6600| 60
1| 4401) +2872] +1283| 19-9627) -7917' 6159) 1
' a1 -4034: -2497| -0899: -9232| -7510: -5740] 2
. 3| 3684 -2139} -0532| -8856] -7122| 5340] 3
| 4 3352 1800} -0185| -8499| -6755| 4961! 4
65 | 3034! -1476| 19-9853] 8158} -6404) 4600 65
| 6] 2733 -1168| 9538] -7835| -6072! -4257| 6
4 | +2446) 0877] -9240| -7529| -5757| -3932) 7
8| -2174, -0599} -8956| -7237| -5457| -3623| 8
9 ue 0334| -8685| -6959| -5171| °3829| 9
vo | 1670! -0086| -8431) -6699| -4903} 3053] 70
1 | °1441| 19-9853} 8194] 6456) 4653) 2795 | 3
| | +1229) 9637} -7974| -6230| 4421, 2555) 2
3} -1034| -9440] -7772| +6023) -4208| -2336) 3
4| 0857] -9259| -7588! 5834; -4014! -2136| 4
vc | -0695| -9095| 7420) 5662) +3837) +1954) 75
6| -0545| -8942| -7265| 5503) +3673] -1785| 6
+1 -0405| -8801} -7121| ‘5355; -3522| +1629] 7
81 -0277| -8671| -6988/ 5220| -3382| 1485; 8
| 9 | 0158] 8551] 6866, 5095; 3253 +1352) 9
| 80} -0050| -8441| -6754 4980 | 3136 | -1231 | 80
' 1119-9952] -8342| -6654| -4877, 3030! -1122| 1
2 | 9864} 8253] +6564) 4785! 2935) +1025) 2
3] -9786| -8174] -6483) -4703; -2851| -0937| 3
4| 9715} -8103 6a 4629, -2774| -0858| 4
"gs | -9650| -8037| -6344; -4560| -2704) -0786| 85 |
6| 9586] -7973| -6279| -4494! -2636| -0716| 6
-| -95294| -7910 6215: -4429 -2569! -0647| 7
| 8] 9463} 7849] 6153, 4365) +2504 0579) 8
'g| -9400| -7786; -6089, -4300, 2437] -0510| 9
igo | 9338 ms 6026 | ‘4236 -2371| -0442) go
—"; i] 9281} +7667; 5970) 4179 =-2312| +0381) 1
| 2} -9231| -7617! +5920) 4128; -2260| -0327| 2
' 3 | -9185| -7572| -5875| -4082| -2213| 0278] 3
' 4] 9148] -7535! -5838 4046 2175, 0240} 4
95 | 9119} 7507] 5810) -4017' -2147| 0210] 95
| 6| -9099| -7487| 5791 3998 | ‘2127' 0190} 6
[175] Age of elder Life:-— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

191

H™.
$i PER-CENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

|

SS | ES | lS | TS eSSS |b ersten sntutie SSeS

53 19-7407 | 19-5692 | 19-3953 | 19-2191 , 19-0398 | 18-8573 | 55
6850| 5116! 3357} -1573|18-9757| -7909| °6
° 6314/ 4561, 2783; 0978; 9140] 7269) 7
8 | -5797| -4027' -2299| -0406| 8548] -6653/ 8
g| 5301} -3514! -1700/18-9858! -7979| -6064} 9
6o | 4825] -3024! -1192 9333 ‘7436 | 5501] 60
i ‘4371| +2554: -0707| -8832! 6917) -4963] 1
2 3937| -2107| -0246| -8355| -6424| -4452| 2
3 | +3525] -1681|18:9806| -7901| 5955; -3966| 3
4 | °3135] +1278; -9390| -7472} 5511) +3507) 4
6s | ‘2762| -0894| -8994| -7062| -5088| -3069} 65
6| 2408} -0529| -8618| -6675| 4688) -2655| 6
4» | -2074| -0184| -8262] -6308| -4309) -2264] 7
8 | -1755|18-9856| -7924| +5959} -3949) 1892! 8
g| 1452] -9544| -7602) -5628| -3607; +1539] 9
vo | ‘1167} -9251} -7300| -5317| -3287) -1208] 70
1} 0902; -8977| -7018/ -5027}) 2987} -0899| 1
2| 0655] -8723| -6756| -4757| -2710| -0613| 2
"3g } 0429] -8490! -6517| -4511] +2456] 0351) 3
4 | 0223! -8278| -6299| 4286] -2225;) 0113] 4
oe | -0035| -8085| -6100) 4081} -2014/| 17-9895 | 75
6 |18-9862| -7906| -5916/ +3892; -1819| -9694| 6
>| ‘9701; -7740| -5745) -3717| -1639| -9508| 7
8 | 9552} -7588| -5588| -3555| +1473] -9337| 8
9 | 9415) +7447) 5443| +3406] -1319| -9179| 9
Bo | -9291} -7318]} -5311| 3271] 1180] -9035] 80
1| -9178} -7202| +5192} -3148| -1053| -8906| 1
2| 9078} -7099| -5085| -3039| -0941| -8790| 2
3 | °8987| -7006| +4990] -2940| -0840| -8686| 3
41 -8906| -6922| 4904] -2852| 0749; 8593) 4
85 | -8831| +6845} -4824/ -2771| -0666| -8507| 85
6| -8759| -6771| -4748] -2692| -0585| -8423] 6
7 | -8688| -6697| -4672| 2614] 0505} -8341]| 7
8] 8618} -6625| -4598| -2538| -0427| -8261| 8
9 | 8547} -6552| -4523| -2461/ -0347| -8179] 9
90 | ‘8476| -6480| 4448) -2385} 0269} 8098} 90
1| -8414] -6415| -4382] -2316; +0199; -8026| 1
2| -°8358| -6357, 4322] -2255| -0136; -7962| 2
3 | 8307} -6305| -4269] -2200| -0080; -:7904] 3
4 | 8267] -6263| -4226] -2156| -0035| -‘7858| 4
95 | ‘8236| -6231| -4193] -2122| -0000| -7822| 95
6| -8215| -6210| -4170| -2099/17-:9977| -7798| 6
[176] Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

192

H™,
$3 PERCENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lxves.
35 36 37 38 39

ee | ery | | | ES | ES | NEES f Me

55 18°6717 | 18-4834 | 18-2925 | 18-0998 | 17-9053 | 17-7089] 5
6025 4114 2175 0216 8236 6236

5
6
; 5361| -3421| -1454|17-9465| -7452| -5416| 7 |
8 | 4721] -2757| -0761| -8743| 6699) 4631] 8 |
g | -4109/ -2121| -0099| -8052| -5979| -3879] 9
60 | 3524] -1512|17-9466| -7393| -5292| -3161| 60
r| 2966] -0932; 8863) -6765| -4637| -2477| 1
2| -2436/ -0381| -8290} -6168| -4015| -1829] 2.
3 | °1932/17:9858} -7746/ 5602) 3425] -1214| 3 |
4| -1456| -9364! -7232| -5068| -2869| -0633} 4 |
65 | -1003| -8894| -6744| -4560| -2340| -0083] 65 |
6 | -0574| -8449) -6282| -4080] -1841| 16-9562] 6
»| 0169} -8029| -5846| -3627) +1370} -9071| 7 |
8 |17-9784| -7630| -5432| -3197| -0923| -s607| 8 |
g| -9419| -7252| -5040| -2790| -0500| -8167| 9
-o | -9076| 6898} -4672| +2409) -0105| -7755 70 |
1| -8757| -6567/ -4330| -2054|16-9736) -7372| 1
2| 8462} -6261| -4013/ -1726| -9395| -7019| 2
3 | -8191| -5981, 3723; -1425/ -9084| -6695| 3
4| °7945| -5727| -3460/ -1163/ -8801| -6402| 4
73 | -7720| -5495| -3220] -0904| 8544] -6135| 75 |
6 | 7513} 5280| -2998| -0675, -8306| -5888| 6
7| -7321| -5082| -2793| -0463| -8087| -5661| 7
8| -7145| -4900| -2605| 0268, -7886| -5452| 8
g| -6982| -4732| -2431] -0088! -7700! -5260| 9
80 | -6833| -4578| -2273| 16-9925] -7531| -5085| 80
1| 6699] -4440| -2130| -9777| -7378| -4927] 1!
2| -6580| -4317| -2002| -9646| -7243| -4787| 2
3 | -6473| -4206| -1888| -9528| 7121] -4661| 3
4| -6377| -4107| -1786| -9422| -7012| -4548) 4
8s | -6288| -4016| -1692| -9325| -6911| -4445| 85
6 | -6202| -3927| -1600| -9231] 6814] -4345| 6
7| 6118] -3840| -1510| -9138| -6719| -4247| 7
8| -6034| -3754| 1422) -9047| -6626| -4150| 8
9 | +5950] -3667| 1332} -8954| -6530} -4052| 9
go| 5867) -3581| -1243| -8863| -6436| -3956| go |
1| -5792| -3604| -1163} -8781| -6352| -3869| 1
2| 5726| -3435| -1092| -8707| -6277| -3792| 2
3) -6666| -3374| -1029| -8641| -6209| -3723| 3
4| -5619| -3324| 0977] 8589] -6154] -3667] 4
95 | -b582| +3287) -0938| -8548| -6112| -3624| 95
6| -5557| -3261| -0911| -8520| -6084| -3594| 6

“e, Joint Life Annuity, at Top.
[177] Age of elder Life:— Last Survivor Annuity, at Side.

198

H®,
33 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
40 41 42 43 44 45

arm er a re | Se | ee | eeeenee | eens eee |

55 17°5105 | 17-3098 | 17-1070 | 16-9024 | 16-6973 | 16-4919 | 5s

4214| +2165} -0093| -7999| -5a98| -3791| °6
° 3357 | -1269|16-9153| -7014| -4864] -2706| 4
8| 2535] -0409| -8252| -6068] -3871| -1663] 8
9 | 1750]/16-9586| -7389| -5163| -2920| -0664| 9

60 -1000 8801 6566 4299 2013 | 15°9710 | 6o
1 0285 8053 5782 3476 "1148 8801 1
2 | 16:9608 "7344 3038 2695 | -0327 ‘7937 | 2
3 8965 6672 4333 "1955 | 15-9549 7119] 3
of "8359 6038 3669 1257 8815 6346 | 4

65 7784 5436 3038 0595 8119 5614 | 65
6 7241 "4869 2443 | 15-9970 7462 49291 6
7 6729 4333 "1882 9381 6842 42701 7
8 6244 3826 "1351 "8823 6257 3653 | 8
9g 5786 3348 0850 8297 5703 3071) 9

70 5357 -2900 0381 “7804 5185 2526 | 70
I “4958 2484 | 15°9945 “347 4704 2019 I
2 *4590 2099 9543 6924 °4260 "1552 | 2
3 4254 1748 9175 6539 *3855 "1125; 3
4 3948 "1430 "8842 6190 -3488 0739 | 4

75 3671 1141 8540 5872 3154 0387 | 75
6 3415 0874 8261 5580 2847 70064} 6
7 3179 0628 8004 5310 -2564|14°9766} 7
8 -2963 0403 ‘7768 5063 2304 94921 8
9 2763 70195 ‘7551 4836 -2065 9241] 9

80 2582 0007 “7354 *4629 "1848 9012} 80
I 2418 | 15-9837 T177 "4443 "1652 8806 I
2 2273 9686 7019 4278 1479 *8624| 2
3 2143 9551 6878 4130 1324 8460] 3
4 2026 9429 6751 3997 71184 8313) 4

85 "1919 9318 6635 3875 "1056 8178 85
6 1815 9210 6523 "3758 0933 8048] 6
7 1714 9105 6413 3643 0812 "7921 7
8 "1615 "9002 6306 3530 0693 ‘T7797 8
9 1514 8897 6197 3416 0574 7670] 9

go 1415 8795 6091 3305 0457 1547 | go
I "1326 “8704 5996 3206 0352 "7437 I
2 1247 8623 5911 3118 0260 7339 | 2
3 "1176 *8550 5836 °3039 0178 T2752) 3
4 "1119 8492 5776 2977 0112 7183) 4

95 1075 8447 5730 2929 0062 7130 | 95
6 1045 8416 5698 -2896 70028 ‘7094 | 6

Joint Life Annuity, at To
[178] Age of of elder Life: - | Last Survi ivor Annuity, at Side.

194

HM,
8} PER-CENT.
Values of Annuities on Two Joint Inves, and on the Last Survivor
of Two Lives.
46 47 48 49 5° 5%

5 | 16-2875 | 16-0848 | 15-8839 | 15-6848 | 15-4878 | 15-2928] sx
6 | -1693|15-9608| -7540| -5487! -3453] -1438| ~6
0553| -8414| -6286| 4173] -2077/14-9996| 7
8 |15-9458| -7264] -5081| -2908/ -0749] -8604| 8
g | -8408} -G161| -3923{ -1693; 14-9473] -7265| 9
60 | -7406| 5107] -2815| -0529) -82951| +5981} 60
r| +6449] -4102| -1758/14-9417] -7083| -4753| 3
2| 5542} -3147| -0754| -8360]} -5971| -3583] 2
3 | -4680| -2241]14-9800] -7357| -4914| 2471] 3
4| +3867] -1384] 8899, 6408] -3915| +1418] 4
65 | -3096| -0573| -8044| +5508] -2967| -04181| 65
6| -2368/14-:9806| -7236| -4657| -2069|13-9472| 6
7} -1681| -9082| -6474] 3853] -1292| -8577| 7
8] +1031] -8398| -5753| -3093| -0420| -7731] 8
9| -0417| -7752| -5072| -2374/ 13-9662] -6930] 9
“o | 14-9844] -7147| -44351 -1703| -8953| -6]81| 70
r| -9310] -6585| -3842] +1078] 8294] -5484/ 1
2] -8818] -6067| -3296| -0502| -7686| -4841| 2
3 | -8369| -5594| -2798|13-9976| -7130| -4254] 3
4| -7962| -5165| -2345| -9499] -6627| 3722] 4
73 | -7591| -4775| -1935| -9066| -6169| -3238| 75
6| -7250| -4416| -1556| -8667| +5748] -2793| 6
71 +6937! -4085| 1208} -8300| -5361| -2383| 7
8 | -6649| -3782| -0889| -7963| -5005| -2008] 8
9| 6384} -3503| 0595] -7654| -4679| -1662| 9
80 | +6143, -3250| -0328| -7372| 4382] +1348] 80
1} -59296/ -3022| 0088} -7119° 4115] -1066] 1
2| -5734' -2819/13-9875| -G894 -3877] -0815| 2
3] ‘5562! -2638| -9684| -6693 -3665| 0591] 3
4| 5407) -2475| 9513] +6513, -3475] -0390| 4
8; | -5265| -2326| -9356| -6348! -3301| -0205| 85
6| -5128| -2182| -9205| -6189| -3133] -0028] 6
7| -4994| -2041] -9056| 6033] -2969|12-9855| 7
8 | -4863/ -1903| -8911] +5880] -2808] -9685| 8
9| +4730! -1763| -8764| -5726| -2646] 9514] 9
go} -4601| -1627| -8621| +5575] -2488| -9347] go
1]. 4484] -1504] 8493] -5440/ -2346] -9198] 1
2| -4381| -1396| -8378| 5320} -2220| -9065| 2
3) -4289| +1299] -8277| -5214] +2108] -8948] 3
4| 4216] -1221| -8195} +5128] -2018| -8853| 4
9s | 4160, -1162] -8133} -5062| -1950] 8781] 95
6| -4121} 1122] -8090{ -5017| -1903| -8732| 6

797 i Age of elder Lifo:_. )Joimt Life Annuity, at Top.

({ Last Survivor Annuity, at Side.

195

H™, .
33 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lrves.

52 53 55 56 57

¢ | 15-1001 | 14-9103 | 14-7243 8:3929 | 8:2240
6 |14:9442| -7475| -5543| 14-3651 O771
7{ -7933| 5896] -3893| -1998| 14-0007

8 | -6474| -4368] -2294] -0256 | 13-8261 | 13-6314
9 | 5069} -2894] -0750/13:8640| -6571] 4548

3720) -1478 | 13-9264 ‘7083 “£940 2843
2429 | 0121 7839 D587 3372 1201
1199 | 13-8827 6478 "4157 1871 | 12°9626
70027; °7594) -5180| -2792 0436 *8120
13°8917 | -6424 "3948 "1494 | 12-9071 6684

‘7863 | 5312 "2775 0258 ‘7769 5314
"6864 "4257 "1663 | 129084} °6531 4010
5919 | -3259 0608 ‘7971 5356 2770
5025 2314 | 12-9609 6914 $239 "1591
“4178 1419 | ° -8662 0912 3180 ‘0471

3386} -0580 7774| 4972) -2185) 11-9418
"2649 | 12-9799 6947 -4096 1256 8435
1968] ‘9078 6183 3286 0398 7525
1347 | -8419 5484 2545 | 11:9612 6691
0783 | -7822 4850] °1871 8897 5932

0271 ‘7278 4273 | °1259| -8247 5241
12-9799 6778 "3742 0694 ‘7647 "4603
9365! -6317 "3252 ‘0175 “7094 4016
* *8967 5894 | +2803 | 11-9697 6586 3476
8600} 5506; 2390} -9258| -6119 "2978

8267 0152 | -2014 8858 | 5693 "2524
‘7968| 4834] -1676 8498 5310] :2116
‘1702 4552 1376 8178 | -4969 1753
7464) -4299 1107 ‘7892 | -4664) -1428
7251 4073 ‘0867 | -7636| -4391 1136

‘7056| +3865] -0645| -7400! -4139| -0868
6868! -3665| -0433| -7174] -3897! -0610
6684} +3470] -0225| -6952. -3661| -0357
6505} -3279| 0022] -6735! +3429] -0110
-6323| -3086|11:9817| -G516 | -3196 | 10-9860

61471} -2899|] -9618] +6304! -2969| -9618
5988} -2731| -9439!] -6114| -2766| -9400
5848] 2583) 9281] -5945| +2586] -9208

5724 2451 “9141 I96 2427 9038
5624 2345 9029 5676 2299 8901

5549 "2265 8943 5585 2202 ‘8797
5496 2209 8884) 5523) :2135 "8726

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2
3
4
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we Joint Life Annuity, at Top.
[180] Age of elder Life:— Last Survivor Annuity, at Side.

196

H™,
34 PER-CENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

7°8645 | 7°6749| 7°4800| 7°2803

"7324 "5502 "3625 "1699

(5935 ‘4187 "2384 5053?

"447 "2805 °3077 "9300

13-2575 "1350| 6'g700 "7999

0793 | 12:8797 "8261 "6637

129076 | -7002 | 12-4988 “3991

-7426 5276 °3185 | 12°1156

5846 3621 "1452 | 11-9346 | 11-7308 3
-4338 2039 | 11-9794 ‘7611 5496 | 11°3451 | 4

2898 | -0525 8206 ‘D947 "3755 1632
"1524 | 11-9080 6688 4354 ‘2086 | 10-9886
0217 7703 5239 2831 "0487 8211
11-8973 6391 3857 1377 | 10°8959 6607
‘7789 | °5142 °2539 | 10-9988 | -7497 5071

CN

6676 3964 1295 ‘8676 6115 3616
0635 2863; :0131 7446 | -4816 2248
4671 1841 | 10-9049 -0303; +3609 0974
3786 0904: -8056 "0251 ‘2498 | 9-9800
2981 0050 | “7181 4292 "1482 8727

™“T

Oo ON AWN PO HHO WYO CK AN

2248 | 109271 6325 3416} :0555 "7746
1570} 8551 D559 2604 | 9-9694 6834
0945 ‘7886 4853] °1854 "8897 3990
‘0370 ‘7274 "£201 1161 8162 5210
10:9840| -6709 3600 0522 7482 "4487

"9357 6194 3051 | 9:9937 6860 3826
8921 5730 2557 9410 6299 "3229
8535 5317 2116 8940 5799 | .:2697
8188 | -4947 1721 8519 °0300| +2218
‘7876 “4615 1367 8141 "4947 1789

7590} -4309 "1040 ‘7792 "4575 1393
‘7314| -4014 0725 7456 4216 "1011
7043) -°3725 0415 7125 3863 0633
6779 3441 0112 6800 "3015 0263
‘6511 3154] 9°9804 °6470| +3162} 8-9885

6252 2876 "9505 6149 2818 "9517
6019 °2625 9236 3860 2508 9185
D813] +2404 8997 5603 2232 “8889
0631 ‘2208 8787 0376 1987 8626
D484 "2051 ‘8617 0194 1790 8413

5373) +1931 8489 0055 1640 "8252
5297 1849 8400 4959 1537 8140

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-e_Ԥ Joint Life Annuity, at Top.
(181) Age of elder Life: Last Survivor Annuity, at Side.

197

H™®,
34 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
64 65 66 67

55 | 68698} 6°6590| 64435} 6225] 5°9057| 5°7611| 55
6| °7731| °5689} °3598| 1450] °9242) °6953| 6
7 6706 "4.732 2707 "0624 8478 *6250| 7
8 "5620| °3716 "1761 9745 "7064} °5500| 8
9 | °4468] -2638] ‘0754) 5°8808| °6795| °4696/ 9
60 | °3260] 1504] 5°9693| °7819| °5876| °3846] 60
I "1998 "0318 8582 6782 "AQLI 2952) I
2 "0686 | 5°9083 "742.3 "5699 | °3902| ‘2016] 2
3 | 5°9329| °7804] 6221 4573 2852} ‘1040] 3
4 6484} °4979| °3408] °1764{ “oo28) 4
65 | 10°9586 *3689 | ‘2197| °0630| 4°8971] 65
6 ‘7762 | 10°5708 70929 | 4°9442] ‘7862; 6
4 6010} °3877 | 10-1816 "8189 | °6689} 7
8 4329 °2119| 9:9978} 9-7908 °5446| 8
9 2717 0430} -8210; -6060| 9:3978 9
70 "1188 | 9°8824] -6527| -4297] -2134; 9-0038/] 70
1 | 9°9748) -7311 4938] -2630/ ‘0388, 88210; 1x
2 °8406| 5898] -3453/ -1070} 8°8750 6493 | 2
3 7167} °4593} -2079| 8-9625| -°7231 "4897 | 3
4 6034} 3398] -0819| -8297| 5834) 3428) 4
75 4997 2303 | 89663) . -7078| 4550) :2075) 75
6 ‘4032{| -1282} °8585| -5940] -3348] -0807) 6
4 3139 | -0337; 7585) -4882| -2230|) 7°9625| 7
8 2312 | 8-946] 6658 | 3900; -1190| 8525) 8
9 1546] ‘8649) 5797) -2988] -0223 7500} 9
80 0844] °7904| 5007) -2150] 7:9334] +6555] 80
I 70209 | -7230| -4291 1390; °8527) -5697| 1
2 | 89643; 6629; -3652; -0711 7805 | 4929] 2
3 9134) 6088; -3076| 0099] °7153| -4236) 3
4 *8677 | ‘5601 -2559 | 77-9547] -6566| +3610] 4
85 8255; 5151 -2080} -9037] -6021; +3028) 85
6 7848| -4718| -1617| -8543| +5494] -2464) 6
7 7446 | 4289] -1160} 8054} -4970; -1903| 7
8 7051 3868} -0710| -7572| -4454|) -1348; 8
9 "6648 | 3438] -0251 7082} ‘3927; ‘O0781) 9
go 6255 | -3019| 7:9804| -6603| +3414] :0227)| go
I 5900} 2640) -9399) 6170] :2949) 6°9724| 1
2 5583 | -2301 9037 | -5783| :2534; 9276; 2
3 ‘5301 ‘2000; -8716) -5440|; °2166; -8879| 3
4 5073} °1756| 8455] -5161] -1867} -8557| 4
95 4900; 1570) -8256| 4948] -1639) 8311] 95
6 4779 | :1440) -8117] -4800; -1481) -8140} 6
[182] Age of elder Life:— er aang ei ae

198

H™.
33 PERCENT.
Values of Annuttics on Two Joint Lives, and on the Last Survivor
of Two Lnves.
70° 71
5 | 5°5216 | 5°3798 55
3 "4613 | °2248 6
7 | °3968 | °1657 »
8 | "3278 | *1025 8
9 | °2539 | °0346 9
60} "1755 | 49625 60
1 | "0929 | °8865 I
2} ‘0063 "8068 3
3 | 4:9160 | “7234 ;
4| °8222 | °6368 4
65 °7 242 "5401 6s
6 "6210 "4 50 5 6
7 | “5117 | °3490 7
8] "3955 | ‘2407 8
Q | °2703 | °1237 9
° 3° 70
— 8-6117 9995 :
2| ‘4319 | 82245 2
3 2646 | -0494 3
4 | °1103 | 7°8876 4
75 | 7:9680 | +7382 75
6| ‘8344 | ‘5977 5
7 | ‘7097 | -4663 7
8! -5933 3435 8
9 | °4847 | -2286 9
80 | °3845 | -1225 80
1 2933 | 0258 I
2| °2116 | 6-9390 3
3 1378 | 8605 3
4| O711 | -7896 4
Bs | -0090| -7234 8s
6 | 69487 | -6590 6
7 8886 | -5946 "
8 |} ‘8290 | -5306 8
g| ‘7678 | +4647 9
go} ‘7079 | -3999 90
1 | °6534 | +3408 t
2| ‘6048 | -2879 3
3 6616 | -2408 3
4| 6266, +2025 4
95 4999 | °1733 95
6 4814 "1530 6
{188} Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

199

Last Survivor Annuity, at Side.

HH,
33 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
76 79 80 81

55 | 41552 | 3°9468 | 3°7412 | 3°5362 | 3°3349 | 3°1422 | 55
6} ‘1209 | ‘or1sg | °7133 | ‘srxx | °3124 | ‘1220] 6
7 | ‘o841 | *8825 | °6831 | °4840 | ‘2881 | ‘1002 ] 7
8 | *o442 | "8464 1} ‘6505 | °4546| ‘2616 | 0765] 8
Q9| ‘0010 | “8072 |} “6150 | °*4226 | ‘2328 ] ‘o505] 9
60 | 3°9549 | °7652 | ‘5770 | °3882 | "2018 |] ‘o225 | 60
I} "9059 | °*7206 | °5365 | °3515 | °1687 | 2°9927 | 1
2} "8542 | °6736| “4937 | °3128 | °1337 | “9611 | 2
3 | ‘8000 |] ‘6241 | 4487 | °2721 | ‘0969 |] °9279] 3
4| °7436| °5726| ‘4019 | *2296] °0585 | °8933 |] 4
65 | °6842 | °5184 | °3526| °1849 | ‘or8s | °8568 | 65
6| *6210| *4607 | ‘3000 }] +1372 | 2°9749 | ‘8178 | 6
7 | °5532 | °3987 | °2435 | °0858 | ‘9283 | °7756| 7
8] "4799 | °3314 | °1820) ‘0298 | °8774| ‘7294 | 8
9| 3992] °2571 | ‘5137 | 2°9673 | °8205 | ‘6776 | 9
70 | °3120| 1764 | °0394] ‘8991 | °7580 | ‘6205 | 7°
I | °2193 | *ogo4 | 2°9598 | ‘8258 | “69061 5586] I
2 | °1230| ‘0008 | °8767 | °7490| “6199 | °4935] 2
3 | ‘0258 | agro1 | °7924 | “6709 | °5477 | °4269] 3
4 | 2°9305 | “Barr | *7096) "5941 | °4767 | °3613 | 4
75. | °8378 | °7346 | “6291 | ‘5194 | °4077 | ‘2976 | 75
6| —— | °6453 | °5458 | °4420] °3360] °2312| 6
7 | 6:4466 "4607 | °3629 | °2625] ‘1631 | 7
8 | :2927 | 6-1175 2810 | °1863 | ‘0923 ] 8
9 | +1476 | 59664 | 5-7949 "1059 | °0173 | 9
80 | ‘0123 | °8255 | ‘6483 | 5-4798 1°9393 | 80
1 | 58884 | -6962 , ‘5136 | -3397 | 5-1764 i
2| ‘7766 | -5795 | :3920} -2131 | -0448 | 4°8894 | 2
3 | °6749 | -4733 | -2812 | -0976 | 4°9248 | -7650/ 3
4| 5827 | -3769 | ‘1806 | 4:°9928 | -8158 | -6519| 4
85 | -4961 | -2863 | -0859 | -8941| ‘7131 | +5454 | 85
6| -4111] +1971 | 49926 | -7966 | -6115 | -4399 | 6
7 | ‘8250 ] -1066] -8975 | -6970| ‘5074 | ‘3316| 7
8 | +2384] -0152| -8012| -5957 | -4012| -2206; 8
9 | °1476 | 4:9191 | -6996 | -4883 | -2880 | -1018| 9
90} -0570| -8229 | ‘5976 -3801 | -1733 | 3:9808 | 90
1 | 49732 | -7337 | :5026 | ‘2789 | ‘0657 | -8668| 1
2} ‘8969 ] -6524 | -4159 | °1862 | 3:°9665 | -7612) 2
3 | ‘8278 | ‘5785 | -3369 |] -1015 | ‘8756 | +6637 | 3
4| ‘7707 | 5174 | ‘2715 | -0311 | ‘7997 | -5820| 4
95 | °7264] -4699 | ‘2205 | 3:9762 | -7403 | ‘5177 | 95
6} +6950] -4862| -1843 -9371 | ‘6979 | 4716] 6

[184] Age of elder Life:— Joint Life Annuity, at Top.

200

H™,
3} PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Inves.
83 84 85 86 87
55 | 2°9037 | 2°7974 | 2°6471 | 2°5063 | 2°3650 | 2.2141 | 55
6 | ‘9456 | "7812 | °6326| "4934 | °3537 | ‘20421 6
7 | "9260 |] °7636| °6169 | °4793 | °3412 | °1934| 7
8 { 9046 | 7444] °5997 | “4639 | “3276 | ‘1816) 8
9 | °8813 | °7234 | “5807 | ‘4469 | °3125 | “1683 } 9
60 | °8564 | *7o0o7 | "5602 | °4285 | ‘2961 | °1540 | 6o
1 | °8292 | °6764 | °5383 | °4088 | *2785 | °1385] 1
2 | *8006| °6506 | ‘5150 | °3878 | °2598 | ‘1220 ] 2
3} 27706 | 6236 | “4906 | "3658 | ‘2401 | “1048 | 3
+] °7394 | °5954 | “4652 | “3430 | ‘2198 | ‘0869 | 4
65 | °7064 | °5656| °4384 | “3190 | ‘1984 |] *0682 | 65
6} °6712 | °5339} “4097 | "2932 | 1756 | *0482) 6
7 "6330 | *4993 "3786 | °2652 | ‘°1507 0265 | 7
8} *591t | ‘4614 | °3442 | ‘2343 "1231 "00241 8
9} 5439 | “4183 | “3050 | "1988 | “0913 | 1°9743 | 9
79 | “4917 | °3706 | *2614 | “1591 | "0555 | "9425 | 70
1] "4349 | °3185 | °2135 | “1153 | "0158 | “Qo71 | 1
2| °3749 | °2633 | °1627 | °0686 | 1'°9732 | °8688 | 2
3 | °3135 | ‘2066 | ‘1¥03 | ‘°o204 | ‘9292 8292 | 3
4 | "2529 | ‘1507 | °0586 | 19728 | °8857 | “7900 | 4
75.\| °1940 | °0963 | *0084 | °*9265 | °8435 | °7521 | 75
6} °1325 | 0393 | 19556 | 8778 | "7989 | “7119 | 6
7 | °0693 | 1°9806 | ‘go1r | °8273 | °7526 | ‘6700 |] 7
8 | -0034| ‘9193 | 8440] “7743 | °7037 | “6257 | 8
9 | 19334 | ‘8540 | °7829 | °7172 | °6508 | °5773 | 9
Bo | °8604 | 7855 | °7186 | .6569 | °5946 | °5256 | 80
1} °7871 | °7166 | °6538 | “5959 | °5375 | 4727] 3
2 "6511 "5919 | °5376 | ‘4828 | ‘4220 ] 2
3 | 46200 "5330 | “4820 | °4306 | °3734 | 3
4 | ‘5033 | 43673 "4326 | *3844 | °3306) 4
85 | +3932 | -2539 | 4:1274 "3426 | ‘2921 | 85
6| 2841 | :1414 | -0117 | 3-8891 "2537 | 6
7 | ‘1718 | :0255 | 3:8924 | -7665 | 3-6410 7
8 | -0565 | 39062 | -7694 |) -6400 | -5107 | 33762 | 8
9 | 39325 | -7774 | -6363 | -5028 | -3691 | °2296 | 9
go} ‘8056 | -6451 | °4992 | -3612 | -2228 | +0779 | go
1} °6855 | ‘5194! :3686 | ‘2261 | -0830 | 2-°9328 | 1
2} ‘5737 | -4018 | :2460 | -0990 | 29515 | -7962) 2
3 | ‘4700 ] :2921 | 1311 | 2:9795 | -8276 | ‘6674! 3
4| °3825 | :1990} -0331 | ‘8773 | °7214 | -5569 | 4
95 | °31383 | °1249 | 29548 | -7952 ! -6358 | -4677 | 95
6| 2634 ‘O711 | "8975 ‘5729 | -4019; 6

[183]

Age of elder Life :—

7351 |

{ Joint Life Annuity, at Top.
( Last Survivor Annuity, at Side.

201

H™,
$1 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lrves.

es | ee | | ES | a. | | Ee. ( S

55 | 2°0534 | 1°8608 | 1°6398 | r°4102 | 1°1725 | o'9130 | 55
6 | ‘ogso | °8538 | *6343 | ‘4060 | "1694 | ‘grog | 6
7 | °0357 | °8462 | °6282 | ‘gorg | ‘1660 { ‘9086] 7
8 | ‘0256 | °8379 | °6216| °3963 | °1623 | ‘go61 |] 8
9} °0143 | °8285 | ‘6141 | °3906| ‘1581 | ‘9033 | 9
fe)

6 ‘oo1g | °8182 | ‘6059 |] °3842 | 1535 | ‘goo1 ! 60
1 | 19886 | “8071 | °5970 | °3773 | 1484 | °8967 | 1
2| "9744 | °7952 | °5874 | °3698 | “1428 | “8929 | 2
31 °9594 |} °7827 | °5773 | °3619 | 1369 | ‘8888 ) 3
4| 9440 | °7698 | *5669 | °3538| ‘1309 | °8847 | 4

65 | ‘9279 | °7564| “5561 | °3454 1 *1247 | “8804 | 65
6} ‘9108 | °7422 | °5447 | °3366|] ‘1182 |] ‘8759 | 6
7 | 8923 | °7268 | °5325 | °3272 | ‘1113, |] °8712 | 7
8 | ‘8716 | *7098 | °§189 | °3168 | +1037 | °8661 | 8
9 | 8474 | °6896 | *5028 | "3045 | ‘0947 | “8600] gy

70 | °*8197 | °6664 | °4841 | ‘2900 | ‘0840 |] °8528 | 70
I | °7887 | ‘6401 | °4627 | °2732 | ‘0715 | ‘8442 | 1
2} °7550 | "6113 | °4399 | 2544] '0574 | "8344 ] 2
3| “7199 | °5812) “4141 | °2345 | ‘0422 | °8237 | 3
4 | °6852 | °5514 |] °3894 | ‘2147 | ‘0270 | ‘8130 |] 4

75 | -6517 | °§229 | °3659 | ‘1960 ] ‘or2g | 8031 | 75
6 | °6161 | °4924 | °3408 | ‘1760 | 0°9977 | °7924 | 6
7 | °5790 | *4606 | °*3146] ‘15521 ‘9819 | °7814 |] 7
8 | 5396] °4267 | °2865 |] °1329 | ‘9650] ‘7696]| 8
9 | 4962} *3891 | "2551 | °1077 | 9458 | °*7561 | 9

80 | *4494 | °3481 | ‘2206 | ‘0796 | ‘9242 | ‘7407 | 80
I "$013 | °3056 | °1844 { ‘0498 | ‘Q008 | °*7239] I
2| °3549 | °2644 | ‘1491 | °0206 | °8778] ‘vo71 |} 2
3 "3103 | °2246 | ‘1147 | 0°9918 | °8548 | ‘6gor | 3
4 | °2712 | *1898 | °0847 | °9667 | ‘8347 | °6752 | 4

85 | °2362 | 1589 | °0583 | “9448 | °8173 | "6624 | 85
6 | ‘2016 | ‘1287 | °0328 | ‘9240 | ‘8009 | ‘6504 | 6
7} *t630 | ‘0951 70046 | ‘QoL! "7831 "6375 | 7
8 0582 | o'9742 | °8771 | *7652 | °6253 | 8
9 | 30841 "9280 | °8393 | °7364 | “6050! 9

go | 29259 | 2-7576 "7849 | °6929 | °5732 | 90
1] ‘7744 , :5975 | 2-4099 | --— 6404 | “S344 | 1
2 | *6317 | :4460 | -2473 | 20512 | --—— | °4867 | 2
3 | 4972 +3030 | -0926 | 1:8828 | 1:6759 3
4+] °3819 | 1802 |} 19591 | -7364 ] -5135 | 1:2874) 4

95 | ‘2892 | -0814 | ‘8511 |] ‘6172 | -3803 | -:1342 | 95
6] 2209 | :0089 | ‘7713 | 5286 | :2809 |] -0179 | 6

‘ Joint Life Annuity, at To

(186) Age of elder Life:— Last Survivor Annuity, at Side.

202

H™,
33 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Inves.

A A | eS | A, |

5 | 06535 | 0°4029 | 01737 | 55
6 | *6522 | ‘$022 } °1735 | 6
7 | “6508 | “4015 | °1732] 7
8 | °6493 | “4007 | "1729|] &
9 | °6475 | 3998) 1726} 9

6o!' *6456 | °3987 ' °1722 | Go
1 | *6434 | 3976 | 74718) 1
2! *6411 | °3964 , ‘1713 | 2
3 | °6386 | 3950 1708 | 3
4 "6360 | °3936| "1703 | 4

65 *6333 | °3922 | ‘1698 | 65
6! +6305 | °3907 1692 | 6
7' +6276 | 3892 | +1686] 7
8 | 6245 | °3876| ‘1680] 8
9 "6207 | “3856 | 1673 | 9

70 | °6163 |} °3833 | ‘1664 |} 7o
I, “6110 | °3805 , "1654 | 1
-2 | °6048 | °3772 , °1642 | 2
3] ‘5980 | °3735 , °1628| 3
4} ‘591r | °3697 | "1613 | 4

75.| °5849 | °3663 | ‘1600 | 75
6| *5780} °3626 ‘1586| 6
7| °5710| °3588! -1571 | 7
8 | 5635 | °3548 : 1550} 8
9 | °555° | °3592 1539 | 9

80 | "5450 | °3448 "1518 | 80
I) °5341 | “3387 | ‘1494| 1
2 | "5231 | °3326 | ‘I471 | 2
3) (5t17 | "3261 | "1445 | 3
4+ "5017 | °3203 | °1422 ) 4

85 | “493t | "3155 | “1402 | 85
6; ‘4851 } “3110; °1385 | 6
7 | *4765 | °3060 | °1364] 7
8 | 4691 | °3021 | "1350 | 8
9: 4503 | "2954 "1325 | 9

9° "4352 | °2835 | "1279 | 90
I "4093 "2688 , ‘1220)| 1
2{ "3776 | ‘251r | ‘IIgr |] 2
3, °3293 | °2228 | °1037-] 3
4 "1839 | °0874 | 4

0:9016 0644 | 9:

*3 7697 | 05260 | 8

te, Joint Life Annuity, at Top.
[187] Age of elder Life:— Last Survivor Annuity, at Side.

eee eee ad

TWO LIVES.

ee
—_—_——

H*,

FOUR PER-CENT.

b bm b=
OO CO™ Aw RPO BD OO

i)

bo SS)
© Or An £$O HHO [HO ON Ah PO WH mw O

fa
re]

fm & DO mw

pf

in
kFO HO HO HO Cr? AN

azz

17°8656

"7937
"6225
"4497
"2539
"043 5
16°8275
6148
"4144
2358
“0809
15°9379
"8006
"6599
*5088

"3455
"1731
14°9931
"8104
6262

“$399
"2522
0598
13°8622
"6585

"4496
"2303
"0260
12°8007
"5779
"3479
"1107
11°8635
6073
"3447
"0760
10°8052
"5339
"2618
9°9873
"7103
"4288
"1412
8°8495
"5557

204

H™,
4 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survwvor
of Two Inves. Equal Ages.

24,—Gz.2

22-9874

2011
"1073
0069
21°9011

‘7903
6761
5992
4414
3238

‘2067
0883
20°9674
"8423
7120

5759
"4345
"2881
1370
19-9816

8219
6574
4876
3122
"1311

18:9440
(513
5528
3487
1384

17-9215
6973
4653
"2257

16-9791

7250
4648
"1987
159262
6473

3617
0688
14-7682
4605
"1461

tn
WO cr~r Qh

N

nN

™“S ~“T!
OO GN QAM FO Ym O OO OT QAn PO HW QO WO GOAN ++ WH BH ™ O

c

CO

Oo
.e]

Ve)
An & O WB Ww

13-8254

"£989
1668
12°8292
4863

"1393
11-7889
"4355
‘0798
10°7228

3635
0019
9°6379
2710
8-9008

-5300
1612
7°7981
4440
1023

67724
4477
1298

5:8177
‘5106

2117
4:9262
6587
4065
1707

3°9424
7092
4603
"1931
2-8880

5521
2045
1°8442
"4996
0727

0-6904
3208

205

H™,
4 PER-CENT.

Values of Annuities on Two Jount Lives, and on the Last Survivor
of Two Lives.

es | ES NS | AS A | GS

17°8x140 | 17°7413 | 47°6511 | 1775476 | 17°4348 | 1

Io oO
1 | 22-2449 6923 6035} ‘5015 "3901 1
2 2001 | 22°1550 "5353 "4348 "3251 2
3 | +1537] -1072 | 22-0575 ‘3009| 2429: 3
4| 1064] -0584| -0076 | 21-9549 1478 4
15 | -0586] 0092] 21:9567| -9023 | 21-8466 as
6 | -0111/ 21-9603] 9062; -8500| -7925/21-7341 6
7 121-9642} -9119] -8563) -7983| -7389| 6787 4
8} -9188| -8649| -8077| -7480! -6868| 6246! 8
9| 8747] -8197} -7608| -6994) -6365] 5725: 9g
20 | -8324| -7758| -7156| -6526| -5879| --5221] 20
1} +7909} 7332) -6714| 6068) 5404] -4728| 1
2} 7501} -6909| -6278; -5615| 4935) -4242: 2
3 | -7093| -6487| -5841| 5163] -4465| 3755! 3
4| 6681} -6064} 5401] -4707| -3993| -3264| 4
23 | -6267| -5635| -4959| -4249, -3516| -2770/ 25
6| -5852| -5208] 4516] -3790 -3039| -2274/] 6
y| 5437] -4781| -4074| -3330| -2564| -1779| 7
8| -5024| -4354) -3634| -2874| -2088) -1286/ 8
9| -4615| -3933| -3197| -2422; +1619) -0798| 9
30 | -4211! -3516| 2765} 1974] -1155| -0315| 30
1| -3808| -3102| 2338] +1530] -0693 | 20-9836 | *
2| +3408] -2689| -1912/ -1089| -0234| -9359| 2
3| -3011| -2280} -1488| -0650|20-9778| -8884| 3
4| -2614] +1873} 1067] 0213} 9325] -8413] 4
35 | -2220| +1466} -0648/ 20-9778] -8873| -7945} 3
6} 1830] 1066) 0233] -9349| -8427| -7480! 6
7; +1443| -0668| 20-9823} -8922' -7985| -7020] 7
8} -1060| -0275| -9416| -8502' -7546| 6565) 8
9 | -0682; 20:9884| -9014/ -8084| -7114] 6114] 9g
40} 0306/ -9499) 8615] -7671| -6685| -5668| 40
1 | 20-9933} -9116] 8220] -7261| -6258] 6225] 1
2] -9562| -8735| -7826| -6854| 5834| -4784| 2
3| 9194] -8359| -7437| -6450| 5416] 4347] 3
4| -8832] -7986| -7055| -6052| -5002]} 3917] 4
4s | -8474| -7619| 6674] 5659] 4592] -3490] 45
6| +8123} -7260| -6303| +5273} -4192] 3073] 6
71 -7779| -6907| -5939| -4896| -3798| -2664] 7
8] -7442| -6562| -5583| -4526] 3414] 2262] 8
9 | -7113| +6224] 5234] -4164| +3036] -1869| 9
so | -6790| 5894] -4893] -3810} -2668] +1484} 50
1| -6475| 5571} 4559! -3463| -2306| -1107|
2| 6165] 5255| -4233|/ .-3124| -1952| -0737| 2
3| +5864; -4946] 3914] -2792| -1606| -0375| 3
4| ‘5571| -4645| 3603] -2469| +1269) -0023| 4

-- , \Joint Life Annuity at Top.
[220] Age of elder Life-— | 1 ost Survivor Annuity, at Side.

206

Last Survivor Annuity, at Side.

H™.
4 PERCENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.
|
16 18 | 19 20 21
tO | 17°3172 | 17°1993 | 17°0856 | 16°9816 | 16°8879 | 16°7987 | 10
Cr} "27391 1575] "O454| 9425] "B504] °7623| 1
2 "21051 °0956] 16°9851! -8839] 7931 -7066| 2
3 "1301 “O170 "9082 | "8087 "7195 63461 3
4 "0368 | 16°9256 8186 | "7208 | °6334| °5502| 4
15 | 16°9346| °8252 7202, *6242 5386 "45721 15
6 "7202 | "OI74 °5230| ° *4392|] 3596] 6
7 | 21°6186 "5137; °4215| 3395] 2617} 7
8 5626 | 21:5012 | °3242 "2440 "1680| 8
9 5086 | 4453 | 21-3835 » "1575 | °0833] 9
20 | 4564] -3913| +3277 / 21-2661 “0086 | 20
I ‘4053 | -3384/ :2730| -2096 | 21:1483 I |
2 3549 | -2861 2189 | -1537{ 0906 | 21-0287) 2
3 3043 | -2337) +1646; 0977] -0328/ 20°9690;| 3
4 2535 | -°1809| 1099} -0411| 20-9744] ‘9088; 4
25 2021 | °1277| -0547 209840; +9154) -8479| 25
6 1506 | -0742|20°9994| -9267| -8561) ‘7867; 6
7 0992} -0209| -9440/ 8694; -7969| ‘7255/| 7
8 ‘0480 | 209677 -8889} -8123| -7379| -6645| 8
9 | 209973] -9151| -8343| -7558) 6794} -6040| 9
30 ‘9470| -8629} -7801| °6997] -6214} 5441] 30
I 8972) -8111| °-7264) 6441] -5639|; -4847/] 1
2 8477 | -7597| +6730; ‘5887| 5067) :4255/] 2
3 7984] -7085| -6199| 5337] -4498) -3667 / 3
4 7493 | 6575) -5671/ °4790| -3932| °3083) 4.
35 ‘7006; -6068) 5146) :4247|) -3371 2502) 35
6 °6524| 5568) 4625' °3708| -2814) +1928) 6
7| 6045} -5071| -4110! -3174] +2263) -1359| 7
8 5572 | -4579| -3601| -2647| -1717| -0795| 8
9 5104) -4093; °3096, -2125| -1179| 0239; 9
| 40 .4639|} +3610; 2596, ‘1607; -0644) 19:°9688| 40
I 4179| -3131} °2099| -1094} -O114| 9142] 1
2 3719) +2655) -1605; -0583])19°9587| 8598) 2
3 3265 | 2182) ‘1116/ ‘0077; -9066| ‘8061; 3
4 2817} 1717; -0632'19:9577; -8551|) -7531] 4
45 2374| -1255) -0155; 9083} 8041} -7007) 45
6 1939} -0804;19:9686) -8599| ‘7543|] -6493| 6
7 1513] -0361| 9227; 8124) °7054| 5990] 7
8 1094199926) -8775| -7658| 6574] +5498] 8
9 0685| 9500} 8334) -7202| -6105| -5015| 9
50 0284) -9082|)+ -7901| +6755) -5645) -4543] 50
1 |19:9890} 8673) -7475, °6316} 5193) -4079] 1
2 9504) 8270) -7059) 5885] -4751| +3625) 2
3 9126| ‘7877| 6650, 5464] -4317| -3181| 3
4 8758 | -7493] -6251| 5052) -3894/ -2747) 4 |
[221] Age of elder Life:— Joint Life Annuity, at Top.

207 °

H™,
4 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
23 24 | 25 26 27

eeneeeneneeend (ene) Ee

|

16°7104 | 16°6183 | 16°5188 | 16-4105 | 16°2951 | 16°1734
6755| ‘§848] -4864) 3796) °2654] +1449
6211] °5319| 4352] °3207] "2171 | ‘0981
"5508 | °4631 °3680| .*2641 153! "0359
"4680 | °3821 "2886 | °1866] *o774) 15°9617

wv ° mt
Oo OTA HPWH HO CO SIAN bBWwOH KH O HO DAM HPO HW =O

1 3767 | °2925} ‘2009; °1006| 15°9933| °8796] 1
2809 | '1986| 1087! ‘or04| “Q050| °7932
1849} “1044 °0165,15°9200| °8166] °7067
"0930] ‘"OI44 15°9284| 8339) °7323| 6245

15°9332 “B4OT | §=—°7565] 6569) “5510

0452 | 19'9529' -8586

20 | 15°9372 | "8621, °7798) <“OBor| 5915} *4875 | 2
"8684 | "7952! °7147 *6259| °5302| 4282
"72931 °6508 "56.40 "4702 °3702
20:9058 "5834 "4986 "4069 "3089
8436 | 20°7781 "4262 "3366! 2408
7807 | 7132 | 20-6449 2883| +1647
7176| +6480! +5776 | 20-5062 ‘0820
6544| -5828| -5102| -4366| 20-3624
5914| 5177; -4429| -3671| -2905 | 20-2137
5290 4531 3762| -2980| -2192| -1398
go | -4670| -3892' -3100| -2295| -1482/ -o6es| 3
‘4055| 3256, -2443| +1615| -0778] 19-9935
3445/ 2624! -1789| -0938| -0076 9208
! 2837| -1996| -1139/ -0264/19-9379| -8485
2233| 1871) 0493 | 19-9595] 8685] -7766
35 | 1633} -0751'19-9851; -8930| -7995} -7051| 36 |
1039] -0137; sae 8272| -7313| -6343

7621 6638 9643
19-9871 8930 7965, -6977 5971 -4950
9296 | :8337: -7352 6341 ‘D311 4266

fe)
I
2
3
4
5
6
7
8
9
°
I
2
3
4
25
6
7
8
9
°
I
2
3
4
5
6
7
8
9

40 | 8728) -7749| -6745| -5713| -4659! -3590| 40
1] -8165} -7167! -6143/ -5090| -4014] -2920| 1
2] -7604} -6590' +5546; -4472| -3374| -2956| 2
3.| -7050| -6019| -4957| 3861] -2741} +1599] 3
4| -6505| -5455| -4375/ -3260| -2118| -0953| 4
45 | 5965} -4900/ 3801; 2667 -1504] -0316] 4s
6| -5438| +4357, -3241] -2086| -0903/18-9694| 6
7| -4920] -3825| -2692| -1520; -0315| -9086| >
8 | -4414| -3303; -2154| -0965/18-9742| -8490] 8
9 | +3918] -2794| +1629; -0422| -9181) -7910| 9 |
50 | °3433| +2295; -1115| 18-9892] -8633| -7342| so
t | 2958) -1806{ -0612| -9373] -8096| -6787] “1
2| -2491/ +1328! -0120| -8865| -7571| -6244] 2
3 | +2036] -0860: 18-9639] -8370| -7059| -5714| 3
4| 1592} -0404 | ‘9171| -7886| -6561/ +5199] 4

“e Joint Life Annuity, at Top.
[222] Age of elder Life: — Last Survivor Annuity, at Side.

208

H™,
4 PERCENT

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

1576505 15°5094 | 1573626 | 1

to | 16°0478 | 15°9189 | 15°7863 fe)
I "0207 "8930 FOI 6270 4872 "3416] 1
2|15°9752| “849t) "71931 °5859} “4474 | °3033| 2
3] ‘9146| *7Q00| 6618) 5301) 3931 | "2505. 3
4| 8424] -7195]} 5929] °4630} 3278; ‘1869, 4.
15 "7620| ‘6410 *5163 *3381 "2547 "1157 «15
6| °6775| °5584| °4357| °3094| "1778| ‘0406 6
7} °5930| °4758] °3550) °2307] “foro! 14°9657 = 7
8 "5127 "39075 2787 "1563 0286! ‘8952: 8
9 "4412 "3279 "2110 "0905 | 14°9548 "83331 9
| 20 "3796| °2683 "1533 "0347 "9108 7812 | 2
! "3223 | °2130] "0999 | 14°9832) 8613 *7336°
| '2663{ 1589] 0479] °9333) 8132] "6875
"2071 "1019 | 149928; 8803 "7024 6387 |

"I1g4i2] ‘0381 °9313| ‘8209] ‘7052
0673 | 14°9566| ‘8621 "7540: °6406
14°9870 "8885 "7865 | "6808 "5699
"9006 | ‘8047 7050]: °
“7171 "6200 "5194 "4136

0

I

2

3

"5837 | 4

5

6

Go1g!] “4935 ;

I
2
3
4
25 "5215 2
6 "$531
7 "3793
8 "3020
g | 20-0605 5318 | 4338 "3307 2218 , 9
30 | 19-9846 | 19-9030 "3447 "2443 | °1382 | 30
I ‘9091 8249 | 19-7410 "1545 ‘o5r3| I
2 8338 | -°7469|] -6603 | 19:5740 13°9595| 2
3] 7589} -6693} -5799| -4907) 19-4014 | 3
4| -6813/ 5920) 4998} -4077| -3154/ 19-2232) 4
3s | -6102| -5151} -4200] -3250{ +2298] -1345! 35
6| 5367] -4390] -3411] -2431] +1448] 0464] 6
+ | -4640} -3635} -2698| -1619| -0605/ 18-9589! 7
8} 3922] +2889] +1854) 0816] 18-9772] 8724) 8
| g | +3212} -2153] +1089} -0021] -8946| “7867, 9
40] -2510| +1424] -0332 | 18-9235 8130 | 7018 | 40
' r| -1814} 0701] 18-9582] -8456] -7320, ‘6177 1
2| -1125/18-9985]} -8838| 7684] -6518; ‘5342; 2
3| 0444} +9278] 8104) -6920] -5724| -4518)| 3
| 4 |18-9773| -8583| -7381] 6169] -4943) “3706: 4
45 | 9113) 7896) -6669| -5429] +4174! -2905 45
6 457 | ‘7227 | -5973| °4707| -3423] -2124 6
7 | -7836| 6572] 5294] -4001| -2690| 1361! 7
8] -7219| 5933} +4630} -3312| -1972/ -0615' 8
9| -6617| 5308} -3982| 2639) -1274|17-9888 9
so | 6029] +4698] -3349| +1982} -0591| 9179 50
1| -5455| 4102] -2731| -1341/17-9995| 8486 1
2} -4892| -3520| -2127} -0714{ -9275| ‘7810 2
3 | 4344] -2952] +1540} -0105| -8642) +7152 3
4| 3810) +2400) 0967} 17-9512], 8027) 6513 4

2 «e. 4 Joint Life Annuity, at Top.
[223] Age of elder Life: { Last Survivor Annuity, at Side.

H™,
4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Laves.
34 35 36 37 38 39
10 as 2099 | 15°0513 | 14°8873 | 14°7186 | ry" 5452 14°3660} I0
I "1899 | °0326 "8696| °7020 "5296 "3517] I
2 15301 14°9969| °8354| ‘“6690] °4980 "3212 | 2
3 "1018 | = "9473 "7872 | ‘6225 "4.528 "27761 3
4} °0398) 8870) °7286| 5654] °3976| °2238| 4
15 | £4°9704 "8192 6627 "5013 "3351 "1632 | 15
6 | 8973] ‘7480| °5932] °4337| 2693} ‘oggr) 6
7 | °8243} °6770) 5240} °3663| "2038) 0354) 7
8 7559| “6101 "4592 | = °3933 "1425 | 13°9760| 8
9 6956} °*5519| ‘4028| 2488] ‘0898 79250] 9
20 6454! °5035| °35621 °2039] 0468) °8836| 20
1} °5990| °4597| ‘3141 | °1636) ‘0083| ‘8469] 1
2 "55551 “4175 ) °2739| °1252/13°9716| “8rar] 2
3 *5088 3728 "2312 "0846 "9328 7751) 3
a "4559 "3221 "1827 "0382 "8886 "73291 4
25 | °3960} 2645] 1273] 13°9850) °8377| *6843 | 25
6 "3301 "2011 "0663 "9264 "7814 "6304| 6
7 "2588 | +1323 "OOO! "8627 "7203 “57171 7
8 1842 | +0603] 13°9308| °7961 6562 "5102! 8
9 | 1067) 13°9856| 8587] +7268] 5897) “4463 9
39 | °0259) °9°77 "7836 | 6545 "5202 "3797 | 3°
11) 13°9419 | 8266) °7055] °5793| 4479] 3104] 1
2 8531 "7407 "62271 “4996! °3712 2368} 2
3; °7588) °6495| 5346) “4147) "2895 | 1582) 3
ne "5524 "4.408 "3241 "2022 0742} 4
35 | 19-0392 "3412 | °2279/ ‘1094/ 12°9848] 35
6 : 18-9478 | 18-8494 "1264| ‘orig | °8903] 6
7! ‘8571] -7553 | 18-6538 12°9085 "7911 | 7
8 | ‘7673! -6621 ‘5571 | 18:4526 *6870| 8
9 | 6783 | -5697 4612] +3530; 18-2454 9
40 5901 ‘47801 -3660) 2542 "1428 | 18-0319 | 40
I | 5026] -3871 2715! 1559) +0407 | 17-9259] 1
21 4159) +2968) +1776] 0583 | 17-9392 *8204| 2
3; °3300] -2075| -0846)17-9616| -8386 ‘7158| 3
4 2455 | #1195 | 17-9931 8662] -7393 *6125| 4
45 1623} -0329 9028 7722 6414 5105 | 45
6 ‘0809 | 17°9482 8145] -6802 5456 ‘41061 6
7 0014; 8654} -7283| 5903 4519 3129] 7
8 117-9238] -7845 *6440| +5025 3603 2174| 8
9 8482 | ‘7057 5618) 4167 2709 1241] 9
50 7743 6287 | -4816 3331 1835 0330 | 50
I) 7022 5536 4033 "2515 0984 | 16°9440) 1
2 6318| -4803} 3268 1718 0152 572] 2 |
3 D634; 4090} -2525 0942 | 16°9343 7727 | 3
4 4969) -3398] -1804; 0190} ‘8558 6907 | 4

[224]

Age of elder Life:—

209

Joint Life Annuity, at Top.
Last Survivor Annuity, at Side.

H®™,
4 PER-CENT.
Values of Annutties on Two Joint Inves, and on the Last Survivor
of Two Lives.
* 40 41 42 43 44 45
10 | 14°1806 | 13°9872 | 13°7847 | 13° 5730 13°3552 | 13°1296| 10
I 1672) 9748) °7733| °5630| °3457| ‘1210] 1
2| 1381] 9469] °7467| °5377]| °3213| ‘og80| 2
3 "0959 | °9062] °7073] ‘4998| ‘2850| ‘o629) 3
4 70437} 8557) °6585| 4524] °2392] °0188) 4
15 | 13°9848] °7984| 6029] °3987 "1871 | 12°9684 | 15
6 | 9226] °7379| °5443| 3418] 1320] ‘g149| 6
7 "8607 | °6779| °4859| °2853 ‘0772 | 8620) 7
8 "8030{| °6220 "4318 2328 70266, ‘8129| 8
9] °7538) °5744) °3859) °1886) 12°9840) °7720) 9
20 | “7I41| 5364) °3495| °1537] 9506] °7402] 20
I 6790] *5029] °3177| °1235 9219} = *7129) 1
2 "64590 | °4715 2880) °0955/] °8954] ‘6880; 2
3 ‘6109 | = =°4384| 2565} °0657 "8675 "6616 | 3
4 *5706| ‘4001 "2202 | ‘0312 "8348! °6308) 4
25 5241 3557 °1779 | 12°QQII "7966! °5945] 25
6 | °4726) °3064| 1308) ‘9462} °7539| °5539| 6
7 "4163 "2526| °0794| °8972 "7072 °5095| 7
8 | °3574| *1963| 0256) ‘8458; °6583} 4629] 8
9 "2962 | 1378] 12°9698| ‘7926] °6075] °4148) 9
gO "2324 °0767 ‘QIN5 | °7370| °5547| 3645] 30
I °1660| °0132 "8508 | .6793 49908 | °3124| I
2} °0954| 12°9457| °7863| 6178) *4413| 2568] 2
3) ‘ozor| 8735) °7174) °5519| °3785| 1972} 3
4 | 12°9394| °7962| °6433) °4813) 3112) °1330) 4
35 | °8535| °7137| °5644| “4058) "2392] *0644| 35
6 "7625 | °6263 "48061 °3257 "1626| 119915] 6
7 | 66 "5345 | °3925| ‘2413) ‘oBar]} 9147) 7
8 "5666! °4380} ‘2999| +1526] 11'°9973| °8338] 8
9} "4605| 3358] ‘2017| 0584) 9071) 7477] 9
40 "2272{ °*0972 | 1t'9581 "B110} °6557| 40
1 | 17-8115 19849 | “Bsor| ‘70731 °5565] 1
2 ‘7019 | 17-5835 | ———— | °7331 "5948 | °4485|] 2
3 5931} -4704) 17:3478 *473 "3320) 3
4 4856 | 3586} -2315 | 17-1048 | "2079| 4
45 8795 | :2480, +1164 | 16-9850 | 16-8545 | 45
6 2753 | 1396) 0033) -8673;| -°7319|16:5974) 6
7 1734} -0333| 16-8926 | -7517|} -6115; 4720; 7
8 0738 | 16-9293} -7840] 6385; -4934/ -°3488/ 8
g | 16-9764] -8277| 6778) -5276| -3776| -2280| 9
50 8813, °7283/ 5740; 4190, -2641 1095 | 50
I 7884} -6311 4724] +3128) -1530/ 15-9933) 1
2 6976! -5363/} -3732{ -2089; -0444; ‘8795| 2
3 76093 | 4439) +2765) -1077/ 15-9383] -7686| 3
4 5236; °3542/ 1826} -0093) -8352) ‘6605/ 4
(225) Age of elder Life:— Joint Life Annuity, at Top.

210

Last Survivor Annuity, at Side.

211

H®™,
, 4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Inves.
46 47 48 49 5° 51
10 | 12°8992 | 12°6649 | 12°4263 | 12°1825 | 1179335 | 11°6778| I0
I *B914| 6580] *4202 "1773 "9290 *6741| 1
2 *B696| °6373| *4006| °1588| ‘9116} °6578| 2
3 "8360| ‘6oso| °3697] ‘1292| ‘88 33 *6308| 3
4| "7933, °5040) °3301| “ogt2) 8467 | = °5957| 4
15 | °7446| °5168} +2847} °0473| ‘B8045] °5550] 15
6 "6929! °4668| °2364| °0006) ‘7594] ‘“5116| 6
7 | 6416) 4172) +1834] 11°9543! °7148) °4685] 7
8} °5943| 3715] 1444] 9118) 6738) *4292| 8
9} °5549| °3337| ‘1080} °8769| 6403] 3970] 9
20 "5245 | °3047|) ‘0804{ "8506 °6153 "3733 | 20
I "4988 | -2804] °0573 *B8289/ 5948 *3540| I
2| °4752| °2583| *0366| “8095! °5767| °3370) 2
3 | “4504| °2349} "0148| *7890! 5576) °3193| 3
4 "$213 | °2075/11'9890| °7648| 5349] ‘2980/] 4
25} °3871) "1750) 9582! = °7358! 5075] *2722) 25
6 "3485 | °1386] °9236| °7030; 4765 "2430| 6
7 | °3062| :0983| °8856| ‘6669| -4424] ‘2107| 7
8 "2620; ‘o564| °8458 "6293! °4068 "1770| 8
9 "2162; ‘o130| ‘B046) “5Q04} “3701 | *1425] 9
30 "1686 | 11°9678| °7619] °5500.; °3320| 1066] 30
I "ITQI | «°9210) °7176) *5082| ‘2926} ‘o6954 1
2 "0664 | ‘8710| °6705 "4636| +2506 0300] 2
3 "0098; 8174} 6197} °4157] 2053 | 10°9874] 3
4 | 119489 | "7597 | °§650|} 3039) 1565] ‘9414) 4
35 8836: .6977/ 5063 *3084/ ‘ro4! "8920 | 35
6 "8143 "6318 "4438 "2493 | *0482 *8393| 6
7 "7412 °5624{ °37791 °18701 10°9893 7837| 7
8 "6641 "4891 *3084 "1271 "9272 “72511 8
9 "5821 “$1II} °2343] ‘0509, ‘8607] 6625] 9
40 | °4944| °3276| 1549] 10°9756| °7894| “5951| 40
1] *3994) 2370) °0687| 8936, -z1t7| “5217|
2 "296! "1381 | 10°9744| °8039: °6264} *4408] 2
3) 1842 "0311 | °8720) °7062 °5335] °3525| 3
4 0650 10°9167 | *7625| *Go16) 4338) °2577) 4
45 | 10°9381| 70948] -6457 4898| °3270| *1560| 45
6 °6670} °5230 "3723 "2147 70489 | 6
7 | 16°3343 "3952 | °2498} 0976] 9°9371| 7
8 2060 | 16-0651 "12181 9°9750| ‘8200| 8
9 0800 | 15-9338 | 15-7895 "B460) °6966| 9
50.1 15°9563 | -8047| 6550 | 15-5073 *5666 | 50
I 8349) -6780| 5228) -3695 | 15-2182 I
2 ‘7159 | 5537) -3930|) °2340| -0768|/14:9216| 2
3 5997; 4321) -2659|) -1012/14-9382| -7770| 3
4 4866] +3136} -1419/14°9716} .-8028) -6355| 4

we Joint Life Annuity, at Top.
[226] Age of elder Life:— } 1 ast Survivor Annuity, at Side,

212

H™,
4 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

52 53 54 55 | 56 57

10 | 13°4147 | 11°1451 | 10°8703 | ro"Syos | 1073065 | 10°0186
I "$116| +1428) ‘8688! °5895) °3061 ‘0189
"3963 | °1285| °8555| °5772| 2948] 0085
3706} = 10gt! §=°83231 5553) °2741 | yp8ge
"3370| ‘0719; ‘8015; ‘5260}] ‘2462 625

"2979 | 0344) °7655| ‘4915] 2132] °9309
"2561 | 10°9942 | °7269| "4544) 1777] “8969
"2147 | = °9543 ‘6886 | °4177/] +1424] °8632
"1767| -9179| °6537] +3842] ‘*1103) °8324
1460} °8884/ °6255| °35731 °0847] ‘8080
"1234| ‘8671 "6053 °3381 "0666 | *7909
1053| ‘8500 "3232| "0526! °7779
0896} °8354| °5757| °“3107| ‘oqro| °7672
°0730| ‘8201 "§616] *2975} ‘o290] °7562
"0531 Bors) "5442 2814] “orgo| °7423

0289] °7787 | °5230) "2615) 9°9954| °7249
"OOlg}| §=6°7529] “4986) °2387}) ‘O740] *7FO50
109709 | °7242| °4716) °2133] "9503| 6827
"9392| *6943| °4436) 1870] 9255) *6595
"9066 | -6637| °4148| ‘1600] *g003) ‘6359
"8729 6319 "3851 "1322 8742 "6117
"8381 | °5993| 3545] “1036| °8476| *5868
"Boog |] 5645} °3219] *0731} ‘8192) *5604
"7609| ‘5270| °2868 “ogog | °7886| °5319
7177| +4864} 2488) 0048] °7554] °5009
6712) 4428) *2079] 9'9565) *7197| 4677
"6217 | = *3963| £643] 9258] °6816| "4322
5693} °3472| 1183] °8828| 6415] °3948
5142) 2954} °0698) 8374] °5993| 3555
"4552| ‘2400] ‘o179} ‘7890| 5540] °3134

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40 | 3918} 1804] 9°9620] °7366} 5052} +2679] 40
I "3224 "IIS! "9007 O791 "4514 "2177 | #1
2 "24.59 "0429 8327 6153 "3915 1617] 2
3 | °1623] 99638} °7581) 5451] °3256] “o999| 3
4 0722 8786} ‘677 "$692 | '2542]| '0330] 4

45 | 9°9757| °7869| *5909| °3873) °1772| 8'9606) 45
6 | °8738| *6903) 4993} “3009| 0957} °8839) 6
7 "7673| °5892| °4036) ‘2103| ‘o103] °*8036| 7
8 6557 °4831 "3030| ‘t152| 89205) ‘“7FIQOT| 8
9} °5380) “371r] "1966| ‘orgq]| “8254] *6294] 9

5° | "4139) °2528| ‘o84r| 8:9078) 7244] °5343 | 50
I "2819 "1268 | 8°9642 "7939 "6165 "$3231 1
2 8°9922 "8358 | "6717 | ‘§006} °3224| 2
3 | 146175 "6994, 5416] °37CQ| "2052] 3
4 | +4698] 14-3065 | “40441 '2462] ‘o8r10| 4

[227] Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

213

, HM™,
4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

58 60 61 62 63
| 10 | 9°7272 | 9°4321 | 9°1350 | 8°8369 | 8°5385 | 8:2409 | 10
| | "7280 | 4335 | °1369 | °8392 | °5413 | ‘2440 ] 1
2| °7186 | °4248 |] ‘r2g1 "8322 | °5350 | °2385 |] 2
3 | °7002 |} °4077 | ‘1130 |] ‘8172 g2ii 22561 3
4 | °6750 | °*3838 | ‘ogo4 7959 | “5011 2068 | 4
15 | °6450 | +3552 | 0633 | “7702 | *4767 | *1837 | 15
| 6 | 6125 |] °3243 | '0338 | °7421 | *4500 1584 | 6
71 *§802 | +2934 | *0043 | "7140 | “4233 | 1330] 7
| B | °5508 | +2653 | 89776 | “6885 | ‘3990 | ‘1099 | 8
9} °5275 | ‘2432 | °9565 | °6685 | “3801 | “og20 | 9
20 | “SII4 |] *2280 | ‘9422 | °6551 | °3675 o8o0r | 20
1} "4993 | °2167 | ‘9318 | ‘6454 | °3585 | ‘o719| 1
7 "4895 | ‘2077 | °9236| °6379 | “3517 | °0657| 2
3|! 4793 | °1985 | ‘or5t | °6303 | *3448 | “O54 | 3
4, °4666} +1867 "9043 6203 3357 0512} 4
25 | “$504 | 1717 | “8904 ] ‘6075 3238 0402 | 25
O! "4317 | "1543 | °8742 | °5924 | “3098 | *0272 | 6
7; ‘4109 | °1347 | 8559] °5754 | ‘2940 | *O125] 7
8 | 3892 | ‘1145 | °8370] °5579 | ‘2777 | 7°9973 | 8
9 | °3672 | ‘og40 | ‘8180 |] “5400 2612 | ‘9821 } 9
39 | °3445 | °0730 | ‘7984 | “5220 | 2443 | 9665 | 30
1} °3215 | °0516} °7786 | °5036 | ‘2274 | ‘9509 ] I
2 | *2969 | ‘0288 | °7575 | ‘4841 | ‘20904 | "9343 | 2
3 | °2704 | ‘oogr | *7347 | °4630} “1900 | ‘9164 | 3
4} 2417 | 89775 | °7099 | ‘4401 | "1688 | “8969 ) 4
35 "2107 | *9487 | °6833 | “4155 | “1461 | ‘8759 | 35
O! 1777 | ‘9181 | *6549 | °3892 | “1218 | °8535] 6
7 °14309 | "8859 | 6251 *3017 "0964 | °8302 |] 7
81 +1065 | 8520 | *5938 | °3328 | ‘0698 | °8057]| 8
9 | *0673 | ‘8158 | 5603 | *3019 | ‘0413 | '77961 9g
40 | "0251 | *7766 | ‘“Sagr | °2685 | ‘or06 | °7513 | 40
1 | 89783 | °7333 | 4839 | °2313 | 7°9793 | “7198 | 1
2| "9261 | ‘6846 | °4388 1895 | °9376| ‘C841 | 2
3; °8683 | °6307 | °3886 | ‘1429 | “8944 | “Oggi | 3
4 | °8056 |: °5721 | *3340 og21 | °8473 | “6005 ]} 4
45 | °7378 | *5086 | °2747 | °0369 | *7960 | °5529 | 45
6; *6659 | “4414 | °2118 | 7°9783 | “7415 | “5024 ] 6
7 | °5905 | *3708 | ‘1460 | “9169 | “6845 | “4494 | 7
8} ‘5111 | '2965 | ‘0766 8522 | 6243 | °3936) 8
9} "4269 | 2175 | ‘0027 | *7833 | *50024 °3341} 9
5° | °3373 | °1334 | 7'9240 | “7099 | “4918 | *2706 | So
Ij °2412 | ‘o431 | °8393 | °6307 | °4179 2019 |} I
2} 1374 | 7°9454 | °7475 | °5447 | °3370 | °1269 | 2
3 | *0265 | ‘8407 | “64G0 | 4522 2510 | ‘o461 | 3
4 | 79088 | °7295 | “5441 | “3536 | “1585 | 019596) 4
[228] Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

214

H™,
4 PER-CENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives. ;

| 64 65 69

i |

7°9447 | 7°6489 | 7°3528 | 7°0554 | 67563 | 645353

| °6528 | "3570 | "0599 | °7610 | “4582
9434 | °6487 | °3536 | °0571 | °7587 | °4565
"9314 | °6377 | °3435 | °°479 | °7505 | °4490
"9138 | *6212 | °3281 | °0337 | °7373 | *4369
*Bg20 | “6007 | °3088 | ‘ors6 | °*7204 | *4212
"8681 | °5780 | °2874 | 6°9954 | “7015 | °4035
8439 | 5551 | °2658 | ‘9750 | 6822 | *3854
"8220 | °5343 | "2461 | "9564 | *6646 | *3689
"Boso | °5183 | °2309 | “O42r | “6512 | °3562

"7939 | °5079 | ‘2212 | "9330 | °6427 | °3484
7863 | *5009 | °2147 | °9271 | °6373 | °3434
"7807 | °4959 | °2103 | "9231 | °6338 | °3403
7751 | °4909 | ‘2058 | ‘grgL | °6303 | *3373
"7676 | *4840 | °1997 | "9136 | °6253 | °3328

\e
I
2
3
4
5
6
7
8
9
Oo
I
2
3
f
5 "7575 | “4748 | “1911 | "9058 | 6183 | °3264
7
8
9
0
I
2
3
4
5
6
7
8
9

an]

wv

"7455 | °4636 | *1809 | "8964 | “6096 | °3185
"7318 | "4510 | “1692 | °8856 | °5997 | “3094
"7178 | °4380 | °1572 1 *8745 | °5895 | *3000
"7036 | “4249 | "1451 | °8634 1 °5793 | °2907

6893 | °4117 | °1330| °8522 | *5690] °2813
“6749 | °3985 | °1208 | “8412 | °5589 | -2721
76596 | °3844 | *1080 | “8294 | °5482 | °2623
"6431 | °3693 | ‘0941 | °8167 | °5366 | ‘2518
"6252 | °3528 | °0789 | °*8028 | °5239 | ‘2402
6059 | °3350 | °0626| °7879} *5103 | °2278
"5853 | °3162 | °0453 | 7721 | *4959 | °2147
"5038 | ‘2965 | 0274 |) °7557 | “4809 | ‘2011
"5414 | *2760 | °0086 | °7387 | °4655 | °1872
"5175 | °2541 | 69887 | °7205 | ‘4490 | °1723

Qs wv
OO G71 An POH HO OHO ON ANN BPW YP HO OHO CON AWN FH DBD HO

so

49 | *4916 | °2304 | ‘9671 | *7oog | °4312 | *1563 | 40
I | *4626 | ‘2039 ] ‘9429 | °6788 | 4112 | 1381 1
2] "4297 | °3736!] ‘9152 | °6535 | °3882] c1r72] 2
3 | °3928 | ‘1397 | “8839 | °6249 | °3620 | °0934 | 3
4} °3525 | ‘1025 | °8497| ‘5934 | °3332 | °0672| 4

45 | °3084 | *0618 | °8122 | “5590 | *3016 | °0383 | 45
6| °2617 | *0186 | °7724 | °5224 | °2682 | °0077] 6
7 | ‘2126 | 69733 | °7308 | °4842 | °2331 | 5°9757| 7
8 | ‘1610 | °9256 | ‘6868 | *4439 | *1962] ‘gqg21 | 8
9 1059 | °8748 | ‘6400 | ‘4009 | ‘1569 ] ‘go62] 9

50 | ‘0470 | °*8204 | “5899 | “3548 | ‘1147 | °8677 | 50
1 | 6°9833 | °7613 | °5354| ‘3047 | ‘0688 | °8258] 1
2{ °9136 | °6968 | °4757 | ‘2497 | ‘0182 | ‘7795 | 2
3 | °8383 | °6268 | *4109 | °1898 | 5°9631 | ‘7290 | 3
4} °7576 | °5517 | °3413 | °1255 | "9038 | °6745] 4

se, _§ Joint Life Annuity, at Top.
[229] Age of elder Life: Last Survivor Annuity, at Side.

215

H™,
4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
7° 7 73
10 | 61498 | 5°8484 | 5°5535 | 5°2691 10
T | '1548 ) °8535 | °5586 | °2743 I
2} °1536| °8528 | 5583 | ‘2743 2
3 | *1469 | “8469 | “5531 | +2698 3
4 | ‘1358 | °8367 | °5438 | 2614 4
15 | °1213 | °8232 | °5314 | *2499 15
6] ‘rog7 | °8078 | ‘sr71 | °2366 6
7 | ‘0877 | °7920 | *5023 | °2228 7
8} 0722 | °7774 | °4887 | ‘2107 " 8
9 | °0603 | *7663 | ‘4783 | ‘2004 9
20 0531 7596 | *4720] ‘1946 20
I} °0485 | °7554| °4683 | ‘1911 I
2] 0458 | °7530 1 °4662 | °1893 2
3 | "0432 | °7508 | °4643 | °1877 3
4] °0392 | °7473 | °4612 | ‘1851 4
25 | °0334| °7421 | °4565 |, *1808 25
6 | 0263 | '7356| 4506 | "1755 6
7 | ‘O179 | °7279 | °4436 | *1691 7
8 | ‘0093 | *7201 | *4365 | °1626 8
9 | °0008 | °7123 | *42904] °1561 9
3° | 5°9922 7045 | °4223 ) °t407 3°
I} °9839 | °6969 | °4154| "1435 I
2} ‘9750 | °6889 | ‘4082 | 1369 2
3 | °9654 | °6802 | *4003 | ‘1298 3
4] "95501 °6707 | °3917 | ‘1220 4
35 | °9437 | °6605 | +3825 | °1137 35
6} °9318 | *6497 | °3727 | ‘1048 6
7 | "9195 | °6386 | °3627 | ‘0958 7
8 | *9068 | *6275 | +3525 | °0866 8
9 | °8935 | °6152 | °3417 | °0770 9
49 | ‘8790 6023 | °3301 | °0666 40
1 | °8627 | *5875 | °3169 | °0547 I
2 | °8437 | ‘5794 | °3014 | ‘o4og 2
3 | ‘8221 | *§508 | °2837 | °0248 3
4] °7982 5292 | °2641 | °0071 4
45} “7719 | °5953 | °2424 | 4°9874 45
©} °7440| “4799 | ‘2193 | "9665 6
7} °7149 | “4535 | °1954| °9448 7
8] +6843 | ‘4257 | ‘1703 | ‘9221 8
9) 6517 | *3961 | °1435 | °8978 9
5° | °6167 | °3644 | ‘1148 | “8719 50
1] °5786 | °3297 | ‘0834 °8435 I
2] °5363 | °2913 | °0485 | ‘8119 2
3 | 4902 | °2493 | °0103 | °7772 3
4 | °4403 | ‘2038 | 479689 | °7396 4

op, Joint Life Annuity, at Top.
[230] Age of elder Life:— Last Survivor Annuity, at Side.

ve

216

H™,
4 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
76 77 78 79 80 81
TO | 4°4939 | 4°2507 | 4°0131 | 3°7788 | 3°5509 | 3°3344 | 10
1 | "4989 | °2557 | ‘0180 | °7836) "5555 | °3388| 1
2! "4998 | ‘2568 | ‘o193 | °7850 | 5570] “3404 |] 2
3} *4969 } °2543 | °0172 | °7833) °5557 | °3393 | 3
4| ‘4907 | :2488| -o123 | -7789 | “5518 | -3359) 4
15 | “4819 | ‘2408 | ‘0050 | °7723 | “5459 | °3305 | 15
6 | "4713 | °2310 | 3°9961 | °7642 | °5384 | °3237 | 6
7 | *4602 | *2207 | "9865 | °7553 | °5303 | °3163 | 7
81 4498 | ‘2110 | °'9775 | °7470] °5226} ‘3091 | 8
9| ‘4418 | ‘2036 | ‘9706 | °7406 |] '5166] +3036] 9
20 | °4372 | “1993 | ‘9666 | °7369 | °5132 | *3005 | 20
1 | "4346 | “1969 | °9644 | ‘7349 | “5514 | ‘2988] 1
2| "4334; ‘t959 | ‘9036) °7342 | *5108 |} °2983 | 2
3 | ‘4325 | *1952 | °9630 | °7338 |] “5105 | *2981 | 3
4 | °4308 | “1937 | ‘9617 | °7327 | °5096| °2974] 4
25 | ‘4278! “1910 | "9593 | °7306 | °5078 | *2958 | 25
6 | 4238 | 1874 | °9561 | °7277 | °5052 | °2935 | 6
7| ‘4190 | ‘1830 | "9521 | *7241 | “5019 | "2905 | 7
8 | ‘4140 | 1785 | “9481 | °7204 | *4986{ ‘2875 |] 8
9} “4091 | "1741 | “9441 | °7168 | "4953 | °2846 | 9
30 | °4043 | "1698 | ‘ogor | °7133 | °4922 | °2817 | 30
I | °3998 | °t656 |] ‘9364 | ‘7100 | ‘4892 | ‘2791 | 1
2 | '3949 | "1613 | °9325 | °7065 | 4861 | °2763 | 2
3] °3897 ; °1566} °9283 | °7027 | °4827 | °2733 | 3
4 | °3839 | "1514 | "9236 | “6985 | °4789 | ‘2699 |] 4
35 | °3777 | *1458 | °9186 | °6039 | °4749 | *2662 | 35
6 | °3712 1) ‘1399 | °9133 | °6892 | ‘4705 | ‘2624 ) 6
7 | *3646 "1339 | "9079 | *6844 | *4663 | ‘2586 | 7
8 | °3579 | “1280 | *9026 | ‘6797 | *4621 2548 | 8
9| "3510! ‘1218 ) “8971 | °6748 | “4578 ) “2510 | 9
40 | 3437) ‘1153 | “8913 | *6697 | *4533 | "2471 | 40
I | 3352 | 1078 | °8847 | °6638 | ‘4481 | ‘2425 ] 1
2| °3251 | *o988 | *8766 | ‘6566 | ‘4417 | °2368 | 2
3| °3134 | °0882 |] ‘8672 | ‘6482 | °4342 | ‘2301 | 3
4 | *3002 | *0764 | °8565 | '6386 | ‘4256 | ‘2225 | 4
45 | °2856 | °0631 | °8446| °6279 | ‘4160 ] °2139 | 45
6| ‘2701 | ‘o4gr | *8320 | *6165 | "4058 | ‘2047 | 6
7 | °2540 | "0346 | ‘8190 | "6049 | ‘3954 | "1953 | 7
8 | +2373 | 0196 | “8054 | *5928 | “3845 | 1857] 8
9 | ‘2195 | "0036 ) "7911 | °5799 | °3731 | °1755 1 Y
50 | ‘2005 | 3°9865 | °7758 | *5663 | “3609 | “1647 | 50
I} °1796 | 9677 | *7590 | °5512 | °3475 | “1527 | 1
2] "1562 | °9466 | ‘7400 ] "5343 | “3324 ] 1392] 2
3 | °1303 9233 | “7199 | °5154] °3155 | “1240 | 3
4 | ‘1022 | "8979 | "6962 | “4949 | "2971 | *1077| 4

Joint Life Annuity, at Top.

[231] Age of elder Life:— Last Survivor Annuity, at Side.

217

H™.
4 PER-CENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

10 | 3°1347 | 2°9495 | 2°7823 | 2°6259 | 2°4698 | 2°3044 | 10
1 | 1389 | "9539 | °7863 | °6297 | °4735 | °3078] 1
2| *1406 '9553 | °*7880 | °6316 | .4753 | *3097 | 2
3} °1398 | °9547 | °7877 | °6314) °4754 “3099 | 3
4) "1368 | *9521 | °7854 | °6295 | *4738 = 3087 | 4

15 | ‘1320 | °9478 ! °7815 | °6260 | °4707 3061 | 15
6 | +1258 | ‘9421 | °7763; °6213 | "4665 -3023 | 6
7 | "1189 | 9358 | *7705 | 6139 | 4616 ) "2979 | 7
8 | +1123 | °9296 | °7648 | ‘6106 | °*4567 , ‘2934 |] 8
9 | ‘107s, "9249 | -7604 | -6065 | -4528| 2899] 9

20 | ‘1042 | °9222 | °7579 | “6042 | °4507 | °2879 | 20
1 | °1027 | °*9208 | *7566 | °6030 | °4496, ‘2869 | I
2| *1023 | °9205 | °7563 | :6028 | ‘4494 | °2868] 2
3 | ‘1022 | *9204} °7564 | ‘6029 | “4496! *2870] 3
4 | "1016 | ‘9199 | *7560 | °6026 | “4494: ‘2869 !] 4

25 | °1002 | °9187 | °*7549 | °6017 4486 °2862 | 25
6| ‘0981 | ‘9169 | °7533 | “6002 | °4473 | °2851 | 6
7| "0955 | ‘9145 | ‘7511 | °5983 | °4450, °2836) 7
8 | °0928 | ‘g120/} °7489 | °5963 | °4438| °2820| 8
9| "Oger | "9096 | "7467 | °5943 | “4421 | “2805 | 9

30 | °0875 | "9073 | °7445 | °5925 | “4404 | *2790 | 30
1 | °0852 | "9052 | °7427 | °5908 | “4389 | ‘2777 1
a | 0827 | -9030| -7408 | “5890 | -4374| -2764| 2
3 | “oBoo | ‘9005 | °7386 | “5871 | "4356 | ‘2749 | 3
4 | ‘0770 |) ‘8978 | °7361 | .5849 | “4337 | ‘2732 |] 4

35 | "0737 | ‘8949 | °7335 | °5825 | ‘4316 | °2713 | 35
6 | 0702 | °8917 | °7307 | °5799 | °4293 | ‘2693 | 6
7 | -0668| -8887 | 7279 | -5775 | 4271 | -2674] 7
8) °0635 | °8857 | °7253 | °575t | “4251 | *2656) 8
g | ‘o6or | °*8828 | °7227 | °5729 | °4231 ! *2639 | 9

40 | °0566| °8797 | ‘7200 | °5705 | ‘4211 | ‘2622 | 40
1} ‘0526 | .8762 | ‘7169 | °*5678 | °4187 | ‘2602 / 1
2] 0476 | °8717 | °7130 | °5644 | “4157 | °2577 | 2
3| °0416 | °8664 | °7082 | “5601 | 4120] ‘2545 1} 3
4 | °0348 | °8603 | °7027 | °5553 | °4077 | ‘2508 | 4

45 | °0270 | °8533 | *6965 | °5497 | “4027 | 2464 | 45
6 | *0188 ; °8459 | °6898 | °5437 | °3974| ‘2418 | 6
7 | ‘orog | °8384 | °6831 | *5377 | °3921 | °2371 | 7
8 | ‘0018 | ‘8307 | °6762 | 5315 | °3867 | ‘2323 ] 8
9 | 2°9927 | °8226 | °6689 | ‘5251 | “3810 | 2274] 9

50 | °9830| ‘8140 | °6613 ! °5183 | *3750] ‘2223 | So
1 | °9724 | ‘8045 | °6529 | *5108 | *3685 | °2166] 1
2| ‘9603 | °7937 | °6433 | °5023 | °3610| ‘2101 | 2
3| °9468 | °7817 | °6325 | *4927 | °3525 | ‘2028] 3

4

4| "9321 | °7685 | °6207 | *4822 | °3433 | °1947

Joint Life Annuity, at Top.

[282] Age of elder Life -— ) Tost Survivor Annuity, at Side.

218

HM™,
4 PERCENT

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

eee ed

Io | 2°1295 | 1°9229 | 1°6885 | 1°4471 | 1.1990 10
I | 1327 | "9257 6o10 | °*4491 2006 I
2; "1345 | °9274 | °6925 | “4505 | °2017 2
3} °1350 |] ‘9280 6931 | *4510 2022 3
4 | “1340, ‘9274 | °6928 | “4509 | *2022 4

15 | °1319 | "9256 | °6915 | ‘4500 | ‘2016 15
6 | ‘1286 9230 | °6894 | °4484 | °2005 6
7| ‘1247 "9197 | °6867 | 4463 | “1990 7
8 | ‘1207 | ‘9162, °6838 | ‘*4440 | °*1972 8
9: *IK75 ; "9133 "6813 | °4419 | *1956 9

20 | °1157 | .g117 | °6799 | “4408 | °1946 20
1} ‘1148 |} ‘g10g | *6792 | “4401 IQ4I I
2| *1147 | °g108 679! "4401 1940 2
3 | “T1149 | “Ori0 194 | °4403 1942 3
4' ‘1t49 | ‘grrr | 6795 | “4404 | 1943 4

25 | "1144 | °Q107 6792 | °4402 1942 25
6 | 1135 | ‘9100 | °6786 | *4398 | ‘1940 6
7! ‘1tat | ‘9089 | °6777 | °4391 | 1935 7
8 | ‘1108 | °9077 6768 | °4384 1929 8
9 | "1094 | "9066 6759 | °4377 1924 9

30 | "1082 | *9055 6750 | °4370 1919 30
1! *ro7r | *9046 6743 | ‘4364 1914 I
2} ‘1059 | °9037 | °6736 | °4359| ‘OIL 2
3. | °1047 | *9027 | 6728) "4353 | *1906 3
4] °1032 | ‘gor5 6718 | +4346 IQOL 4

35 | ‘1016 | “goor | °6707 | °4337 | °1895 35
6} ‘0999 | °8986 6695 | °4328 1888 6
7 | *0982 | 8972 6684 | °4319 1881 7
8 | *0967 | “8959 | °6673 | 4310] +1875 8
9 | "0952 | *8948 | °6664 | *4303 | °1869 9

40 | °0938 | *8937 | °6655 | ‘4297 | °1865 40
r | °0922 | °8924 6646 | °4290 1860 i
2 | ‘ogor | °8907 6633 | ‘4280 1853 2
3 | °0874] °8885 |} .6616 | °4267 | ‘1844 3
4| ‘0842 |] 8859 | °6595 | “4252 | °1833 4

45 | °0805 | °8828 | ‘6570 |] °4232 | °1819 45
6 | 0766 | *8794 | °6543 | ‘g2tr | °1803 6.
7 | °0724 | °8761 | °6516 | ‘4190 | °1787 7
8 | °0684 | °8727 | °6489 | "4169 | °1772 8
9 | *0642 | 8692 | 6461 | “4148 | °1756 9

go | °0598| -8657| °6433 | *4127 | “1740 50
rj} °0551 | °8618 | °6403 | ‘4103 | ‘1724 I
2{| °0495 | °8572 | °6367 | °4076| °1704 2
3] °0433 | °8521 | °6326 | ‘4045 | *168r 3
4 | °0364 | "8464 | °6281 | “gor0 | 1656 4

te Joint Life Annuity, at Top.
[238] Age of elder Life:— Last Survivor Anwuity. at Side.

219

H™,
4 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lnves.

96
10 0°4084 | 0°1757 | 10
I “4088 "1759 I
2 "4091 "1760 2
3 "4093 | ‘1761 | 3
4 "4093 | “1761 |) 4
15 "4.093 "1761 | 15
6 "4091 *1760 6
7 "4088 | "1759 | 7
8 "4083 ‘1758 8
9 4079 | *1756] 9
20 "4076 "1755 | 20
I “$075 "17 54 I
2 "$074 | °1754 2
3 "$075 "1754 3
4 4075 1754 | 4
25 "4075 *1754 | 25
6 4075 | “1754 ] 6
7 "4074 1754 7
8 "4072 | 1753 | 8
9 “$071 "1753 9
30 °4070 *1753 | 3°
I "4.069 "1752 I
2 *4068 "1752 2
3 "$007 | “1752 | 3
4 "4006 | “1751 | 4
35 24065 “E752 1 35
6 "4063 "1750 6
7 "4062 | "1749 / 7
8 "4060 | ‘1749 ] 8 a
9 "4059 "1748 9
40 "4058 | *1748 | 40
i 4057 | ‘1748 |) 1
2 4056) "1747 | 2
3 "4054 | °1740 | 3
4 "4052 | 1746] 4
45 “$049 "1745 | 45
6 "$045 | °1743 | 6
7 “fO4T “1742 7
8 "4038 | ‘1741 8
9 "4034 "1739 | 9
50 "4031 "1738 | 50
I "4027 "1737 1
2 "4023 °1735 2
3 "4018 | °1733 | 3
4 "4013 °1731 4

op. Joint Life Annuity, at Top.
[234] Age of elder Life:— Last Survivor Annuity, at Side.

220

H™,
4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
10 II 12 13 14 15

55 20-5286 | 20°4355 | 20-3303 | 20-2156 | 20-0941 | 19-9680 | i 55
0011 ‘4074 3012 1853 0624 9348 | 6

’ 4746 3802 ‘2731 1560 0317 9027 7
8 4489 3540 2459 1277 0021 ‘8715 8
9

g| -4242| -3987] 2199] -1004119-9735| 8415
60| -4005| -3045| -1948| 0743; 9461] 8126! 60
1| 3779} -2815] -1710| -o494! -9199] -7850j 1
2 | -3564| -2595| -1483/ -0256| -8948| -7586} 2
3 | -3357| -2385] -1265| -0028| -8708] +7333! 3
4| 3163] -2187| -1060/ 19-9814] -8482] -7094) 4
6s | +2979] -1999| -o865]} -9609| -8266| -6865| 65
6| -2804| -1e21} -ocao|- -9415! -8061| -6648| 6
7| 2640) +1654) 0507] -9233| -7867| 6442, 7
8 | -2483| -1495] -0343] -9059; -7683| -6246| 8
g| -2336] -1346] -0188/ -8897| -7510] -6061| 9
-o| -2197] -1206| 0043] -8744] -7347| 58861 70
1| -2068] -1076]19-9908| -8601! -7195| -5724! ‘1
2! 1949] -0957| -9785| -8471! 7056] 5574/2
3 | -1840| -0847| -9672| -9351| -6997| 5436] 3
4| 1741] -0749| -9570| -8244/ -6812} ‘5311! 4
<5 | -1651| 0659] -9477] -8145, -6706| 5197] 75
6| -1568| -0577| -9393| -8056 -6610| -5092| 6
»| -1493| -0502| -9316] -7975: -6522| -4996| 7
8| -1493| -0433] -9245| -7900! -6441) 4908] 8
9| 1359] 0370) -9181| 7832} -6368| 4828; 9
80} +1299] -0312| -9122} -7769. -6300| -4753' 80
:| -1245| -0260| -9069| -7714' -6240| -4688, 1
2} -1198| -0215| -9023| -7665' -6187| -4629/ 2
3 -1156| -0174| -8989| -7622° -6140| 4577 3
4{ 1118] -0137| -8945| -7582| -6097| -4530 4
ss | -1084| -o105| -s911| +7547! -6058| -4487) 8s
6 1051] -0073| -8880 7481 | 6021 4446 6
»| -1018| -0043| 849] -7481/ -3985| -4405! 4
8} -0986| -0013| -8220| -7449! -5951| 4366! 8
9 | -0952/19-9983| -8791 7419 | 5917| -4329| 9
go| -0919| 9953] -8763| -7391; 5886] -4293| go
1| 0887] -9926] -8737| -7366| -5859| -4262, “x
2{| -0858| -9901] -8715| -7344| 5836] -4236! 2
3} 0830} -9878| -8695| -7395' 5816, -4213. 3
4| 0806] -9858| -8678/ 7309 5800 4196 4
95| 0787) -9842/ 8664) -7206, 5788. 4182. 95

‘0774 "9831 8655 °7288 | ‘0780 | 4174 | 6

“e Jvint Life Annuity, at Top.
[205] Age of elder Life:— Last Survivor Annuity, at Side.

221

H™,
4 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

16 17 18 20 21

2325

| | 19-8400 | 19-7119 | 19-5863 | 19-4651 | 19-3483 | 19: 55
6| -8052| -6757| -5487| -4262| -3083| -1916| 6
>| -7716| -6405| -5122| -3885/ -2696| -1519] 7
8 | -7389| -6064| -4767| -3519| -2320| -1134] 8
g| -7073| -5734| -4424] -3164| +1956] -0762| 9
60 | -6770| -5417| -4093| -2823] +1606] -0403| 60

|r| -6480| -5113| 3777] 2496] -1270| -0060| 1

' a} -6202] -4821| 3473] 2181] -0947]18-9730| 2

' 3] -5935| -4541] -3181] -1879| -0638] -9413] 3
4 | 5682] -4276| -2904| 1593] -0344| 9113] 4
6c | -5441| -4022| -2639] -1318] -0062] -8825| 6s

5211| °3779| +2385] -1056/18-9793| 8551! 6
-4993| -3549| -2144| -0806| 9537] -8289| ;

| 4784| 3329; +1914] 0567] -9292] -8039| 8

: 4587| 3120] -1694| 0340] +9058] -7801/ 9g

7 44014 2923] +1487} 0125} 8837} -7576 | 70

4927 "2737 "1292 | 18-9922 8629 ‘7364
-4066 2566 ill ‘9734 8437 ‘7167
3918 2408 0944 9560 8258 6986
3783 2264 0792 9402 8096 6820

3660 2132 | -0652 ‘9256 "T94T 6668
"3547 2010 0523 9122 7808 6527
3443 1898 | -0404 8997 ‘7680 6397
3346 1794 0293; °8881 756) 6276
3258 1699 0191 8774 7451 6164

3177 1610 0096 "8675 "7349 6060
3105 1531 | *-0012 8586 "7257 5967
3040 "1461 | 18-9936 8507 “7176 "5884
2983 "1398 “9869 8435 7102 “5809
2931 134] "9807 8370 ‘7035 74]

"2883 "1289 ‘9751 8311 “6974, -5679
2837 1238 9696 8254 6915 5619
"2792 1188 9642 8196 6856 5559
2748 "1139 9588 | -8139 6797 5499
2704 1089 9533 | 8081 6737 5438

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I 2627 1000| = -9432|) -7972 6623 5323] 7
2 2596 0963 ‘9390| °7925 6575 5273 | 2
3 2569 0932 9354] °7884 6531 0228 | 3
4 "2549 0907 9324) °7851 6495 5191) 4
95 2533 ‘0888 9302 7825 6468} 5162/95
6 2524 0877 9287 ‘7808 6449 5143 | 6

en

Joint Life Annuity, at Top.

(206) Age of elder Life:— Last Survivor Annuity, at Side.

222

H®,
4 PERCENT

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

55 19-1159 | 18-9962 | 18-8716 | 18-7418 | 186077 | 18-4699
0741
0335
8 | 18-9941
9| 9561

9194
8844
8507
8184
‘7878

‘7584
7304
‘7038
6783
6541

6312
6097
5897
5713
5545

0391
5248
5116
4993
‘4880
"A775
4681
"4597
4521
"4453

4390
"4330
4269
-4209
"4148

4088
4032
"3983
3938
"3901

3872
"3852

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3
4
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3
4
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6
7
8
9

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of, Joint Life Annuity, at Top.
[207] Age of elder Life:— Last Survivor Annuity, at Side.

223

. H™,
4 PERCENT.
Values of Annuities on Two Jot Inves, and on the Last Survivor
_ of Two Lrves.
28 30 3 32
55 18:3293 | 18°1865 | 18-0413 | 17-8938 | 17-7432 | 17: 5894 5

2793 1347 | 17-9878] -8383] -6856
, 2309 0847 | :9359 ‘7847 | -6300
8 1841 0363 | -8860| -°7329 5764
9 "1390 | 17°9897 8377 6830 | °5247

6o ‘0957 9449 | -7915|) 6352 ‘4752

OO CON Aw

Lo,
oO

I 0541 9022 ‘T7472 5895 "4279
2 0144 "8611 ‘7050 | ‘5458 "3827
3 | 17-9765 8219 6645 5040 3395
4 "9404 7848 ‘6261 "4644 2986
6 "9060 7493 | 5895 4266 2596

8732 7155 5546 3907 2224
"8421 "6834 5216} 3565 "1872
8123 6527 4900 | -3240 1536 | 16:9787
“7841 76236 | -4600] -2931 "1218

0 wx ad fO& Wb

5
6
/
8
9
7° “7574 5961 4317 2639 "0917 7°
I 7323 5703}; -4051 2366 ‘0635 I
2 ‘7091 5464 3805} = :2113 ‘0374 2
3 6877 0244 3578 1879 0134 3
4 6683 | 5044) +3372 1667 | 16-9916 4
75 6504 $860} +3183) 1472 9715 15
6 6339 4690) 3008] -1292 "9530 6
7 "6187 4533) +2846) -1127 9359 7
8 6045 4387 2697 0973 "9201 8
9 0915 4253 | -2558} -0830| 9054 9
80 0794} 4129 2430} -0699 8919 80
I 5686 "4017 2316 0581 8798 I
2 5589 3918 2214 ‘0476 8690 2
3 5903 3829 2122} 0382 8593 3
4 5424 3748 2039 0297 "8505 4
85 5352 0674] +1962 0218 8425 85
6 5283 3602 1889 | -0143 "8347 6
7 5214] 3531 1816 0068} °8270 7
8 5145 3461 1743 | 16-9993 8194 8
9 5076 3389 "1670| -9918] -8116 9
go 5008 "3319 1598 9844) -8040 go
I "4946 3255 1532} ‘9777 ‘7971 I
2 4891 3198 1473 9717] -7909 2
3 "4842 3148 1421 9663] -7855 3
4 "4802 3106 "1378 9619] -7810 4
95 4771 3074 1345 9585 | -7775 95
6 "4750 3052 1322 9562 7751 6

-- {Joint Life Annuity, at Top.
[208] Age of elder Life: Last Survivor Annuity, at Side.

224

H™,
4 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
34 35 36 37 38 39
55 17-4326 | 17-2729 | 17-1106 | 16-9462 | 16-7799 | 16-6113 | s5¢
3705| 2082 -0433] -8760| -7065! 5348] 6
° ‘3106| +1458! 16:9783| -8083] +6359] -4610] 7
8 | -2527| -0857| -9157| -7430| -5678| -3900| 8
g| -1971| -0279! -8555] -6803| -5025| -3217| -9
6o | -1439]16-9725,; -7979| 6203] -4399| -2564] 60
1|/ -0930| -9196; -7429! -5630| -3802| -1941] 1
2| 0444] -8691 | 69041 5084} -3233] -1348] 2
3116-9980] -8210! -6404| -4563] -2691| -0782] 3
4 | -9541| -7754| -5930| -4071| -2178| -0247| 4
6s | -9123] -7321| 8479} -3602| -1690| 15-9739 | 65
6| -87261 -6909| -50521 -3157| -1228{/ -9257] 6
71 -8349| -6518! -4646| +2736] 0789] -8801] 7
8| -7990| -6146| -4260| -2336| -0373| -8368] 8
9} -7650| +5794] -3895! -1957|15-°9979| -7958] 9
7o| 7328} -5461) 3550, +1599] -9609| = -7572] 70
1| -7028] -5150' -3298! -1265| -9261| -7212| 1
2| -6750| -4862: -2930!/ -0956| -8941| -6879| 2
3| -6494]} -4597| -2656| -0672| -8647| -6573| 3
41 -6261| -4357; -2407| -0414] -8380| -6296| 4
vs | +6048 aast| ‘2179| -0178| -8135| -6043| 75
6| -5851} -3933; -1968'15:9960| -7910| -5809} 6
»| -5669| -3745| 1774] 9760] -7702] -5594| 7
8 {| -5501]} -3571| -1594| :9574| -7510; -5395] 8
g| -3345/ -3411 | 1428] -9402) -7332| 6211] 9
80] 5202} +3262: -1276| +9244] -7169| +5042] 80
I 50731} -3130' -1188} 9102} -7093| -4891] 1
2{ -4958| -3011! -1016| -8976| -6892| -4756| 2
3| 4856] 2905; -0907| -8863| -6776| -4635|] 3
4| -4763] -2809| -0807] -8761} -6670| -4526] 4
8s | -4677| -2721| -o717| -8667| 6574] -4426| 85
6| -4595| -2636| -0629| -8577| -6480| -4330] 6
7 | -4513| -2552}) -0542| -8487] -6388| -4235| 7
8 | -4432] -2468] -0455!/ 8398] -6296| -4141] 8
9] -4349| -2383| -0368| -8308| -6204| -4045| 9
90 | 4269} -2300} -0282| -8219| -6113| -3952) go
r| -4195| -2224| -0203} -8138| -6030| -3867| 1
2| 4130] -2156{ -0133/ -8066| -5955| -3791] 2
31 -4072| -2096| 0071! -8002| +5889} -3723| 3
41 -4024/ -2047| -0020| -7949| -5835| -3667| 4
95 | 3988] -2009|15-9981| -7908| +5793] -3624| 95
6| -3963| 1983} 9954] -7881] 5764| -3595| 6
[209] Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity, at Side.

225

H*™,
‘4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor

of Two Lives.

40 41 42 43 44 45

55 16-4407 | 16-2675 | 16-0917 | 15-9140 | 15-7353 | 15-5558
3606 | 1837} 0040} -8220| -6388| °4544}+
2835 1030 | 15-9194] -7333 5456 | 3566
2092} -0253|} -8379| 6478] °4559] 2623
1379 | 15-9505 7596 | -5656 3696 | °1717

0696; ‘8791 6846} °4869 2869 0848
0045 8110} 6132) 4119; :2081 0019
15:9425 | . °7461 5452} +3405 "1330 | 14°9229
"8835 6843 | °4804) 2725 0615| -8477
*8276| 6259; °4192] -2082)14:9939| -7766

“7746 5704} -3611 "1471 9297 | -7090
‘7243| °5178| -3059{ -0893; 8689 "6450
"6767 "4681 2538; 0345) -8114| -5844
6316 | °4209 2043 | 14°9826| 7568] 5270
5888} 3763] -°1576 "9335 ‘7051 4726

487} °3343| °-1137| 8874] 6567) -4216
‘5111 -2952| -0727 8444] -6114} 3739
4765) 2590] 0349 8047} -5697] °3300
"4447 2259] -0001 7683 | 5314] +2897
‘4159{ +1958] 14:°9687| °7352] °4967 2532

3896 1683] :9400| -7051 "4651 °2199
3652| 1430; -9135 6773} 4359 "1891
3429 | +1197 8891 6518] +4090 "1609
3223| 0982) 8667 6282 | :3843 1348
3032 0784 8460 6065 3615 1108

2857 | 0602] °8270| 5866] -3406;/ -0888
2700} 0439} °8100; 5688] -3218| -0690
2561 0294| 7948) 5529) -3051 0515
2436 0164} °7813| 5387) -2902| -0358
23231) -0047 7690} 5259] -2768; 0216

2220 | 149940} -7578) 5142] -2644| 0086
-2120| -9837| -°7471 5029} °2526| 13-9962
‘2022! 9735) °7364| 4917) :2408| -9838
1925| -9634) °7259} -4807| :2293/ -9716
1826 | -9532 ‘7153| :4696; :2176| 9593

"1731 9433| -7050| 4588} -2063 9474
1643 | 9343] -6957 "4491 1960} °9366
1565 ‘9263; °6874| 4404; -1869 9269
"1496| -9192]) 6800) -4328/ 1789) -9184
1439 9134) 6740) 4265) 1723] 9115

1395 9089} 6694) 4217) 1673] :9062
1365| °9058) 6663) 4185} -1639} -9026

in
© CO~ Qn

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OW ONAN BOK HO /O CON

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I
2
3
4
5
6
7
8
9
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I
2
3
4
5
6
7
8
9

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oe Joint Life Annuity, at Top.
[210} Age of elder Life :— Last Survivor Annuity, at Side.

226

H™,
4 PER-CENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lnves.

wn
© o~! Aun

2704 0871 14: 9046 7230 5427 3634
"1678 | 14:9794} -7916 6046 4184 2332
0687 8754] “6825 4900 2983 1072
149734] -7753| 5773 3796 1824 | 13-9855

8822 6793} 4764 2736 0710! -8685
‘7950 | =°5877 "3801 1723 | 13-9644 | -7564
‘7119 5002] 2881 ‘0755 8626 6493
6327 4170} +2005 | 13-9833 7655 5470
‘5578 3382) ‘1175 8959 6735 “4500

"4867 2633 0387 8128 6859 3578
"4193 1922 | 13-9639 7340 5028 ‘2701
3555 1250} -8930 6593 ‘4241 1870
2949 0613 8259 5885 3494 1081
2377 0010} -7623 5215 2787 0334

1840 | 13-9444 7027 4586 2123 | 12-9632
1338 8915 6470 3999 1503 8978
0876 8428 5956 3457 0931 "8373
0451 ‘7981 5485 2961 0407 7819
0066 ‘7576 5058 2510 | 12-9931 7316

13-9715 7206 4669 2099 9497 "6857
9391 6865 ‘4309| -1720} -9097 6434
9094] +6552 3979] 1372 8730} -6046
"8819 6262 | -3675 1051 8391 5687
8567 5996 3394] -0756; -8079 5358

8335 5752 3138 0485 7794] -5056
8127 5534 2907 0242 “7537 "4785
7942 5339 2702 0026 7310 4544
‘T777 0165 2519 | 12-9833 7106 "4329
7628 5008 | -2354 9660} 6923) -4135

“7491 4864 2203 :9500| -6755 "3958
"7360 4726] +2057 "9347 76594 "3787
7229 "4589 1914 9196 6434] -3619
‘7101 "4455 "1772 9047 "6278 3453
6972 "4318 1629 8897 6119 3286

6846 4186 | 1490 "8751 5966 3124
6732 4066 1364 "8618 5826 2978
6630 3959 1251 *8500 5703 2847
6540 "3864 1151 8395 5593 2732
6467 3787 1070; -8310 5504 2638

6411 3728 "1008; :8245 5435 2567
6373 3687 0965 8200} 5388 2517

°,)
ve)

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~

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6
7
8
9
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2
3
4
5
6
7
8
9

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-- ‘| Joint Life Annuity, at Top.
[211] Age of elder Life: Last Survivor Annuity, at Side.

H®,
4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.
52 53 54 | 55 57

55 | 14:3256 | 14-1560 | 13-9891 8-1081 | 7°9496| 55
6 1852 0092 8358 | 13°6656 "B115| 6
0490 | 13°8665 6866 5097 | 13:3363 7
8 |13:9169; -7281 5417 "3581 "1779 | 13-0015 | 8
9 ‘7891 5941 4012 2110 0240 | 12°8406| 9
60 *6662 4650 2658 0689 | 12°8751 *6849 | 60
I 5483 3411 1356 | 12°9323 7318 5347 | 1
2 4355 "2224 0108 ‘8011 5941 3902} 2
3 3279 "1090 | 12°8914 6754 4620 2515 | 3
mr "2256 0012 ‘7778 5558 3360 1190} 4
65 1282 | 12:8985 "6695 4416 2157 | 11°9923 | 65
6 0357 8008 5663 3328 "1009 8713) 6
7 |12:9479 7081 4683 2292 | 11-9916 ‘7560 | 7
8 8646 6200 3752 1307 8875 -6460| 8
9 7856 5364] 2868 0372 7886 5414] 9
rk) ‘7114 4578 2036 | 11:9491 6953 4427 | 70
I 6421 3844 "1258 "8667 -6080 3502] 1
2 5781 3166 0539 "7905 "5272 "2645 | 2
3 5194 2544 | 11:9879 7205 4529 1857 | 3
4 4661 1979 9280 6568 3853 "1140; 4
75 4176 1464 8733 “988 3237 0485 | 75
6 3727 0989} = -8229 5452 2667 | 10-9879 | 6
7 3316 0552 "7765 4958 2142 9321) 7
8 2936 0149 7336 4503 "1658 8806 | 8
9 2586 | 11-9778 6942 4083 1211 83301 9
80 2266 "9438 6581 3699 0803 ‘7895 | 80
I "1979 9133 6256 3354 0435 ‘7503 | 1
2 1724 8862 5968 3047 0108 7155/ 2
3 "1496 8619 ‘5710 2772 | 10-9815 6842] 3
4 "1290 8401 5478 2525 "9551 76561) 4
85 1102 8201 5265 2298 "9309 6303 | 85
6 0921 8009 5060 "2080 ‘9076 6054; 6
7 0743 ‘7819 4859 "1865 8847 5809} 7
8 | -0568| -7633| -4661] -1654| -8622| -5568| 8
9 0391 "7445 4461 144] "8394 5324) 9
go 0219 7263 "4267 1234 8173 5089 | go
I 0064 7098 4092 1047 7974 4875] 1
2 | 11-9926 6952 3936 0881 7797 4686; 2
3 9804 6823 3799 0735 7641 4519] 3
4 9705 6718 3687 0616 7514 4384) 4
95 9630 6638 3602 0525 ‘7417 42801 95
6 "9578 6583 3544 0463 ‘7351 4209] 6

[212] Age of elder Life:— } Joint Life Annuity, at Top.

227

Last Survivor Annuity, at Side.

H®™,
4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Inves.
58
55.| 7°7841| 7°6114] 7°4327| 7°2486| 70599] 6'8673 | 55
6 "6528 "4869 "3150 "13761 6°9554 "7692| 6
7) °5148) 3559) "1908) "0203) “8449] °6653) 7
8 2181 "0601 | 6°8966 7282 *5554| 8
9 | 12°6613 69224 "7661 6047 4390] 9
60 4985 | 12°3164 76296| 4755] °3169} 60
I 3413! -1520| 11-9677 -3409| 1895] 13
2 "1898 | 11-9935} -8019 | 11-6158 70572] 2
3 | 0443] -8409| -6422| -4489] 11-2613 3
41119052] -6949] -4891| -2886| -0937]10-9049| 4
6s | -7719| -8548] 3421] +1345] 10-9324] -7363] 65
6] -6444| -4207| -2011|10-9865| -7773| °5739| 6
+1 +§927| -2995| -0662| -8447| -6284] -4178| 7
8 | -4067| -1700| 10-9371] -7088| -4856| -2679| 8
9! -2961] 0532] -8139] -5789| -3488] -1241| 9
70 | -1917110-9428| -6972] -4558| -2190] 9-9875| 70
I 0937! -8391| -6876| -3399/ -0968] -8586| 1
2{ -0030| -7430] -4857| -2322! 99899] -7385| 2
3|10°9194] -6544] -3919| +1328] -8778| -6275!| 3
4| -8433| 5736} -3062) 0421] -7817| 5259) 4
75 | -7787| -4997| -2278] 9-9590] -6937] -4327| 75
6] -7093! -4313| -1551| -8818} -6119}] -3460| 6
71 -6499| -8681| -0880| -8104| -5362] -2657/ 7
8 | -5951| -3097| 0258] -7444| -4660| -1913] 8
9| -5445| -2558!] 99684] -6832| -4010{ -1222| 9
80} -4981| -2063| -9157]) -6271] -3413] -0588) 80
I -4563| -1618| -8682| +5765} -2874| 0014] 1
2| -4192| -1292|) -8259] -5315| -2394| 8-9503| 2
3 | -3858| -0866| -7879| -4909} -1962] -9043! 3
4| +3558] -0546| -7538| -4545| -1573] -8629] 4
85 3282! 0251] -7223! -4209| -1215) -8247)] 85
61 -3017| 9-9967| -6919]| -3885! -0869] -7879| 6
7 | -9754| -9686] -6619| -3564| -05296| -7513] 7
8 2496| 9410] -6323] +3247! -0188] -°7152| 8
9 | +2236] +9130] -6023| -29296] 89844] -6784| 9
9°} +1983] -8859| -5732] -2613| -9509| -6426] go
I 1754| +8613] -5468| -2329] -9204] -6099| 1
2| -1552| -8395| -5233| -2077] -8933] -5808| 2
3 | +1373] -8203| -5026| -1854} -8692] -5549| 3
4] -1228| -8047| -4858| -1673| -8497] 5339] 4
95} -1117| 7928] -4731| +1535] +8348] 51781 98
6} -1041| -7846/ -4642| -1439| -8245] -5067| 6
[213] Age of elder Life:— Joint Life Annuity, at Top.

228

Last Survivor Annuity, at Side.

229

H™, ;
4 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lnves.

64

6°6713 | 6°4713
"5796 | °3857
"4822 | °2947
*3789 "1980
"2694 | *0953

"1544 | 5°9872
"0342 | °8741
5°9092 | °7563
7797 | =°634
*5080

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10°5468
3771 | 10-1864
2138 0157
‘0566 | 9°8512
9:9057 6929

"7620 5420
6263 3994
4947 2661
3826 "1426
2753 0294

1768 | 8:9254
0850 8283
0000 7383
8°9211 6547
"8478 6771

‘7804 5056
‘7195 4409
6652 3831
6162 3310
5722 "2841

5315 2408
4922 1989
4532 1574
4148 1164
3755 0745

3372 0336
3023 | 7:9964
‘2711 "9631
2434 9335
"2209 9093

2036 8908
1916 8780

“nN

© Or An HOP ew QO LW COn~ Qn

7

6
7
8
9
)
I
2
3
4
5
6
7
8
9

~

“I

©
or a ee re)
c
Oo

co
ioe)

Oo on A
WO CO™ QA PLD

‘oO
o
Ve)
Oo

\O
Amr & © WP mm

Oo
On &O DD mM

| «- ¢ {Joint Life Annuity, at Top.
[214] Age of elder Life:— Last Survivor Annuity, at Side.

230

H™,
4 PERCENT.

Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lnves.

70 71 72 73 74

55 5°1546 4°6987 | 4°4827 55
6 ‘1018 8758 "6548 | °4427 6
7 "0452 | * "6076 | *3996 7
8 4°9846 | ° "5508 | °3532 8
9 ‘9194 | * "5020 | = °3031 9

60 Bsor | ° "4437 | °23497 60
1 “7771 , *3821 "1931 I
2 "7003 . "3172 "1336 2
3 *6202 | ° "2402 | ‘O7IL 3
4 "5309 | ° "1785 | ‘0061 4

65 "4496 | ° "1043 | 3°9378 65
6 3575 | °° 0257 | °8654 6
7 "2597 | ° 3°9418 | 7879 7
8 1553] ° 8518 | *7045 8
9 "0423 | 3° "7536 | *6133 9

70 3°9222 | ° "6487 | °5154 7°
I 5379 | *4118 I
2 7°9834 "4238 | °3048 2
3 "8174 "1970 3
4 6638 | ° 7:2765 4

75 ‘D216 | ° 1209 | 6:9404 75
6 ‘3877 |: 69739 | -:7871 6
7 2624 | -042 8359 | -6429 7
8 1449 | 6: ‘7060 | 5071 8
9 0349 | : 5839 | 3791 9

80 6°9328 | : 4703 | °2599 80
I 8399 | - 3665 | +1509 I
2 ‘7564 | 5082 | -2730 | -0526 2
3 “6808 | °42% 1882 | 5:9633 3
4 6124; -: "1114 | °8823 4

85 5485 | :287 0395 | -8065 85
6 ‘4863 | -2213 | 5-9692 | -7323 6
7 4239 | - 8984 | -6572 7
8 3616) : 8273 | °5817 8
9 2974 | : ‘7533 | -5030 9

go 2341 | 5°9507 | -6800 | -4246 go
I 1762 | - 6123 | +3521 I
2 1241 | - ‘S511 | :2863 2
3 O777 |: -4960 | ‘2269 3
4 0399 | -739% 4506 | °1778 4

95 0109 | - 4157 | -1398 95
6 59907 | - 3912 | °1130 6

“¢., Joint Life Annuity, at Top.
(215) Age of elder Life:— te Last Survivor Annuity, at Side.

eee -. tte, —-_ ae _

231

~ H®™,
4 PER-CENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lnves.
78 719 80 81
55 3°6712 | 3°4725 | 3°2770 | 3°0896 | 55
6 6442 | ‘4482 | *2551 0700 | 6
7 6150 | *4219 2315 0488 | 7
8 5834 | °3933 | °2058| -o257 | 8
9 5490 3622 1778 0004 | 9
60 512] 3288 | °1476 | 2°9732 | 60
I 4728 | °2933 | “1155 | "9442] !
2 4313 2556 | ‘0814 |] -9134 | 2
3 3877 | °2161 | 0456) ‘8811 | 3
4 3423 | °1749 | ‘0084 | -8474 | 4
65 2945 | ‘1314 | 2°9690 | +8118 | 65
6 2435 o851 | *9271 7739 | 46
’] 1886 0352 | ‘8818 7328 | 7
8 1289 | 2°9807 | °8322 6879 | 8
9 0625 | "9200 | °7768 | -6373| 9
70 2°9902 8535 | °7159 5816 | 70
I 127 7820 | ‘6502 5212 I
2 8317 7071 “5811 4576 | 2
3 7495 | °6309 | *5106 | °3925 | 3
4 6687 5560 4413 3284 | 4
75 5902 | *4831 | °3739 | °2661 | 75
6 5089 4075 | °3037 2012 | 6
7 4258 | °3301 | “2319 | "1345 |] 7
8 2501 | ‘1574 0652 | 8
9 5:6670 0786 | 19917 | 9
80 5258 | 5°3639 "9152 | 80
I 3961 | °2289 | 50715 I
2 2788 1067 | 4°9445 | 4-7946 | 2
3 1719 | 4:°9953 | -8286 | -6744}; 3
4 0747 8940 7233 | -5651 | 4
85 4:9833 7986 | -6240 | +4622 | 85
6 8930 7043 | °5256 | -3600 | 6
1] 8008 6076 | -4246 | -2548] 7
8 7070 5090 | °3212 | :‘1468]{ 8
9 6079 4043 | 2108 | 0308} 9
go 5082 2984 | -0986 | 39125 | go
I 4150 1992 | 3°9931 8006 | 1
2 3297 1080 | -8956 6967 | 2
3 2519 0246 | -8059 6007 | 3
4 1872 | 3°9549 | -7309 5199 | 4
95 1366 9005 | :6720 4561 | 95
6 1006 8617 | °6298 4103 | 6
[216) Age of elder Life:— Joint Life Annuity, at Top.

Last Survivor Annuity at Side.

232

H™,
4 PERCENT.

Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

84 85

2°6077 | 2°4706
"5936 | 4580
"5782 | 4442
"5614 | ‘4292
"5428 | “4125

"5228 | °3945
"5014 | = °3752
4787 | °3547
"4548 | °3332
"4299 | °3108
"4038 | ‘2873
"3757 | °2622
"3453 | °2348
"B1I7 | °2045
‘2734 | °1698
"2308 | °1309
*1839 | ‘0880
"1341 | *0422
70828 | 1°9949
"0322 | °9482

"9829 | °*9028
°9312 | “8550
8778 | °8055
8218 | “7534
‘7618 | *6974
*6986 | °6381
"6349 | °5780
"5741 | °5207
"5161 | °4660
—— | *4174

4-0580

3°9459 | 38274
8301 | °7083
‘7103 | 5851
5804 | 4511

4463 | °3126
3180 | °1799
"1975 | -0549
0842 | 2:9371
2°9873 | °8360

9096 | -7546
8527 | :6949

Ww

Oo Cn Qn
WW
wy

O COn~r A

“n
@)
“v
oO

f & WB
& OG BO

“
n
CA

ay
~Y
nrPw ww OO OO COn~ OA

~
~

5
6
7.
8
9
o
I
2
3
4
5
6
7
8
9

oO

CO
© Cr~ Ain PH WO ww O
we He OOWaA

fe)
WO co~r Qn A

‘Oo

©
fh © PB wm O

gO BO tO

\o

\o
On
On

we Joint Life Annuity, at Top.
[217] Age of elder Life :— oe Survivor Annuity, at Side.

233

H™,
| 4 PERCENT.
Values of Annuities on Two Joint Lives, and on the Last Survivor
of Two Lives.

tn
A nw end 6 BAN

0 6) Osan
“WN
© OX Qn O&O WO

~
~

wT
~
© O~7 Aw Aw O = O

I
2
3
4
5
6
7
8
9

c

h& OF Y HO
co

[o °]
b> ww BW mm O

© Or Qn
co

WO fn Qn

‘Oo

An POD HO
Le)

Oo DH & O

\o
‘6
nw

e,. {Joint Life Annuity, at Top.
(218) Age of elder Life:— } J ast Survivor Annuity, at Side.

234

H®™,
4 PERCENT.
Values of Annutties on Two Joint Lives, and on the Last Survivor
of Two Lives.

55 | 0°6493 | 0°4007 | 0°1729 | 55
6 | ‘6480 | ‘4000 | '1726] 6
7 | °6466} °3993 | °1724| 7
8 | °6451 | °3985 | ‘1721 | 8
9} 6434 | °3976] 1718} 9

60} ‘6415 | °3965 | °1714 | 60
1 6393 | °3954| ‘1710] 1
2 | °6370] °3942| ‘1705 | 2
3 | °6345 | “3929 | ‘1700 ] 3
4| °0319 | °3915 | ‘1695| 4

65 | *6293 | “3901 | °1689 | 65
6 | *6265 | °3886 | +1684] 6
7 | °6236| °3870 | °1678] 7
8 | ‘6205 |] °3854] ‘1672 !] 8
9 | "6168 | °3835 | ‘16651 9

70 | ‘6124 |] °3812] +1656 | 70
rj} 6071 |] °3784| ‘1646] 1
2| ‘6010 |] °3751 | *1634| 2

| 3] °5943 | °3715 | ‘1620] 3
4} °5874} °3677 | 1605] 4

75 | “5812 | °3643 | 1592 | 75
6 | °5744] °3606| 1578] 6
7| °5674 | °3508 | *1564) 7
8} *5600} 3529] 1549] 8
9) “5510 | °3483 | "1531 | 9

80 5417 | °3429 |] ‘1511 |] 80
1} °5308 | °3369 | ‘1487 | 1
2] °5199 | °3308 | °1464] 2
3 | *5086 | °3243 | °1438] 3
4} "4986 | °3186} ‘1415 | 4

85 | “4901 | °3138 | +1395 | 85
6 | *4822 | °3093 | ‘1378 | 6
7| °4737 | °3044| °13581 7
8 | °4663 | °*3005 | °1344] 8
9| ‘4536 | *2938| ‘1319 | 9

9°} °4326| °2820] °1273 | go
t | ‘4069 | °2673 | ‘1214] 1
2| °3754} °2498 | ‘1146 ] 2
3 | °3274 | °2216} ‘1032 | 3
4 "1830 | °0870] 4

0°8959 "0641
°6 | -7579 | 05931 c
. Joint Life Annuity, at Top.
[219] Age of elder Life:— Last Survivor Annuity, at Side.

AUXILIARY TABLES.

236

HX,
Auxiliary Series, for the formation of Survivorshi
Y Son the Jorma e P

Lo
* (p3!—1)

A —A

_ Log
A aCe! +1)

10 | 3°692 329

re]
Amn b& 0 mH

; > &o &o 1) SN)
Awe mH OO OO GO Qh Ph Pm O WO COr~ 7 OK ho Ww GO WO CO~

aN
in

i
POH moO OHO O~ AH

*602 660
"522 599
"465973
"442 351
‘459 280
°512 870
"59° 772

682 605
"761 796

"804 096
"830620
838 305
"833 160
"825022

*824 406
828 022
"842 132

"858 747

"874 424

891 135
"902 034

"O12 282

"921 gI0

"933 317

"947 008

"963 423
*980 190

"994 627

2°007 929

7017 580
02.5 220
"03.5 465
"O51 232
"067 875
“ogi 408

‘117 600
"142 784
"165 877
"189 076

"200 751
"220 234
"251950
277 678
"303 874

910 331
919 849
943 404
976 378
016.929

053 59°
077 902
091 833
079 191
O42 300
026 524
007 685
994 855
991 862
999 384

003 616
Ol4 110
016 615
035 677

O16 711

010 899
010 248
009 628
OIFI 407
013 691

O16 415
016 767
O14 437
O13 302
009 651

007 640
O10 245
org 767
016 643

023 533

026 192
025 184
023 093
023 199
020 675

O19 483
022 716
025 728
026 196
028 270

089 669 | o'001 068

o80 151

056 536
023 622
983 071
946 410
922 098
908 167
920 809
957 79°

973 476
992 315
005 145
008 138
000 616

996 384
985 890
983 385
984 323
983 289
989 101
989 752
990 372
988 593
986 309

983 585
983 233
985 563
986 698
99° 349

992 360
989 755
984 233
983 357
976 467

973 808
974 816
976 907
976 8o1
979 325

980 517
977 284
974272
973 804
971 730

"0900 869
"000 722
"000 634
"000 601

"000 625
"000 706
"000 846°
"OO! O45
"O01 253

"001 381
"001 468
"O01 495
"OO! 476
"001 449

"OO! 447
"OOT 459
"OO! 507
"001 566
"001 623

"001 686
"007 730
"O01 771
‘oo! 811

“oor 859

"001 918
"O01 92
"002 070
"002 139
"002 207

"002 256
"002 296
"002 350
"002 436
002 532

"002 671
"002 838
"003 007
"003 17)
"003 343
003 505
"003 666
"003 861
"004 096
"004 349

999 801 | 000 199
999 853 | 000 147

999 O12
999 967
000 O24
000 081
000 140
000 199
000 208
000 128

000 087
000 027

999 981
999 973
999 998
000 OI2
000 048
000 059

000 088

000 O19
000 027
000 002

999 988
999 952
999 941

000 057 1999 943
000 063 1999 937

000 044 | 999 956
000 O4 I | 999 959
000 O40 | 999 960
000 048 | 999 952
000 059 | 999 94!

000 074 | 999 926
000 078 | 999 922
000 069 |990 931
000 068 | 999 932

000 049/999 951

000 040 | 999 960
@00054 999 946
000 086 |999 914
000 096 999 904
000 139 | 999 861

000 167 '999 833
000 169 999 831
000 164 999 836
000 172 '999 828
000 162 | 999 838

000 161 ' 999 839
000 195 | 999 805
000 235 | 999 765
000 253 1999 747
000 292 999 708

mh

© Bb HO

So OMY On >

SJ

>]
0 OO OF AN PO WY

&o oso
% OO OF Qn PO HN HF

— GO Bb

>

a
he O WO Orr Qn

Log
* | (ps!—1)

2°332 144
301 135
"39° 573
"419951
452 071
"485 512
"519 812
"554 839
"599 512
"624 407
"657 080
688 806
"720 228
"749 947
"784 103
"821 620
"863 437
"908 544
"95591

T'0OI 502
"037 791
"075 671
"112413
"147 762
°186.057

228 181
"273 462
315 518
358 468
"394 885

"424 293
"449 454
"478 246
"497 728
°530 252
*588 644
"658058
"733 O61
"852 284
0°'012 681

"244 302
"647 817

A

028 gg!
029 438
029 378
032 120

033 441

034 300
035 027
035 673
033 895
032 673
031 726
031 422
029 719
034156
037 517
041 817
045 107
047 367
O45 591
036 289
037 880
036 742
035 349
038 295

H*™,
Auatiary Series the formation of Survivorshi;
ary eos Ser Tabhe eontisnedy af

971 009 | 0'004 641

963 711
962 120
963 258
964 651
961 705

042 1241957 876

045 281
042 056
042950
036 417
029 408
025 161
028 792
o19 482
032 524
058 392
069 414
075 603
118 623

160 397
231 621

403 515

"O04 959
"005 306
"005 674
006 105
"006 591
"007 128
"007 722
008 377
"009 050

"O09 749

"010.479
O11. 255

°012 O41
"013 O12
"014 167
"O85 573
"O17 244
"019 189
"021 261

"023 065
"025 108
027 256
"029 491
"O32 113

"035 253
"038 957
"042 730
"046 940
050814
"054 161
"057 188
"060 84.5
"063 442
068 008

"076974
"089 030
"104 073
"132 208
"180 359

"273 592
°§08 155

000 318
000 347
000 368
000 431
000 486

000 537

000 594
000 655

000 673
000 699

000 730
000 776
000 786
000 971
OO1 155

OO 406
ool 671
C01 O45
002 072
oo1 804

002 043
002 148
002 235
002 620
003 142

003 704
003 773
00.4 210
003 874
003 347
003 027
003 657
002 597
004 566
008 966

012056
O15 043
028 135
o48 151
093 233

234 563

A

999 682
999 653
999 632
999 569
999 514

999 463
999 406
999 345
999 327
999 301

999 270
999 224
999 214
999 029
998 845

998 594
998 329
998 055
997 928
998 196

997 957
997 852
997 795
997 380
996 858

996 296
996 227
995 79°
996 126

996 653

996 973
996 343
997 403
995 434
991 034

987 944
984 957
971 865
951 849
906 767

795 437

tn
WO Or~? An

3

oO now wn

~

Ve) Cc foo) ~J
hw Pm O © CO~ Cin BO HY mw OO LO CDW CQin A&® BHO Oo com OQ

\O
Qn

"029126

966184 "033816
961538 "038462
959938 | °043062

"047619
"056604

END OF TABLES.

Logarithms.

1-987163
985060
982967
980884
‘978811

974694

2-464284
529128
‘585027
634096
677781

‘752845

ay

NOTATION.

a aaa Ege peenponnstensn nenamyatnerntensnnenpensnnaseennsoatpemememmamemennseeed
Se

NOTATION.

interest on £1 (or any other monetary unit, say
on 1,) for one year.

a = the value of 1 due a year hence,
L+¢

log .(1+4)=Nap Log (1+1)== the force of discount ;

the nominal yearly rate of interest when (interest
being convertible momently) the effective rate of
interest is $.

the effective rate of interest when the nominal rate
of interest ¢ is convertible m times a year.

Do. do. half-yearly.
Do. do. quarterly.
Do. do. momently.

the number of persons living at the age z according
to any mortality table.

N.B. Although the subscript form of notation is used here and
throughout this scheme, it will be seen that the notation can equally
well be adopted in its essential features by those who may prefer the
functional form,

Ux), &e.
l.,,== the number of persons living at the age z+ 1.
bein ” ” r+n.
1p \= ” ” z—l1,
LL .»= ” ” T— Ne

1,—/,,,== the number of persons dying between the
ages z and z+1.

dryly — Z42

d rinlbein— zZ+n+l

dz\=lz_\— lz

drip = ten—lr—n4t

Lily loys = daly ls

bp telyst Mews = best lys idea
beiybay Neg = be hay de)

bey— Vy=lply —_ beat bt

241

"ry — desnyin =", — —*t],, — een lytn— lope lent

"dey = dz» y—n= "ley AT len ly en antl tant

n~m).. —= be tmath 8" -=Ozin—m:

N.B. When there is no risk of misunderstanding, the subscript
2, zy, &c. may be omitted. Thus in lieu of

_ le, giles oe
aT +0 I, +o - + wc cee
we may write
yy et
—=er +e 7 + 7 + 2. ce eee °
la ad ‘ty at th
He =— 7" Bem _ Jy Bs l.=the force of mortality at the age #.

We have approximately

ee es dz_)+dz
Pe Wu, .w

w (omega) = the limiting age, which none of the lives actually
attain.

the probability of a person whose age is 2 living
years,
the probability of a person whose age is 2 living one
year.
the probability of the joint existence of two (n years,
persons whose ages are z and y vontinting| 1 year.
nPxy) = the probability of the joint existence of (n years,
three lives z, y, 2, continuing {i year.
aPsy) = the probability of the existence of the last (n years,
survivor of z and y continuing tt year.
= the probability of one at least of x, y, and z, being
alive at the end of » years.
nPzy;2 = the probability of two at least of x, y, and z, being
alive at the end of n years.
\= the probability of 2 dying within f ‘ rear.
ad == the probability of the joint existence of 2 {i years,

nPx

AY
ll

Bs

a
ye”
Il

_ and y failing within I year.
== the probability of the last survivor of 2(n years,

and y dying within 1 year.

cs
a
woe

= the probability of z dying before y mat ye
I

242

m'ngx == tho probability of z dying after the lapse of m years
but before the end of the following n years.
N.B. In this expression m and n may be either integral or frac-
tional.
== the probability of x dying within a year, deferred
n—119z n—1 years ;
== the probability of + dying in the nth year from’ the
present time,
a-ifzy == the probability of the joint existence of and y
failing in the nth year.
n-1l%zj == the probability of the existence of the last survivor
of z and y failing in the nth year. ;
a-1191y = the probability of x dying in the nth year and leaving
y surviving.
Qi, = the probability of x dying before y.
6, = the curtate “expectation,” or average duration, of
life, of a person of the age =z.
@zy = the curtate expectation of the joint existence of two
persons of the ages z and y.
== the curtate expectation of the existence of the last
survivor of z and y.
ls@x == the curtate expectation of life during the next n

years of a person of the age z.
nlég == the curtate expectation of life after the next n years
of the life x.
have corresponding meanings for the joint existence
ntay ‘al of z and y, and the existence of the last survivor,
tay olay respectively.

The complete expectation is denoted by é: thus

e e ° e °
Cx, Oxy, Oxy 3 LsOxy nlx 9 &c.

Endowments.
als == the value of an endowment of 1 payable in n years
if x be then alive.
ad = Do if { and y hare alive.
nlizys x, y, and z
ad = Do. _ if one at least of {: and y his alive.
allzys 2, y, and 2

sE—3 = Do. _ if two at least of 2, y, and x are alive.

243

Annuities.
@z = the present value of an annuity (or annual payment)
of 1 due at the end of every year through which
z shall live.

N.B. It is always understood (unless otherwise stated) that the
annual payment of-the annuity is 1.

Obs. An annuity is to be conceived as accruing (or growing)
during each instant of the life, lives, or status, on which it depends;
but as becoming payable only yearly, half-yearly, or at other stated
intervals during the year. In the ordinary case no payment is
made on account of the annuity for the year, half-year, quarter,
or other interval, in which the life drops or the status fails, When
@ proportionate part of the annuity is payable for the portion of a
year, half-year, or other interval, which has elapsed from the date
of the last payment to the day of death, or failure of the status; or, in
other words, when the annuity is payable up to the day of death; the
annuity is called complete.

Ae+1

Gz—)

GAzrin-m

the value of an annuity payable during the joint
existence of 2 and y.

ditto ditto of z, y, and z.

ditto ditto of x, y, 2, w, &.

Gz+l-y+1

Age\yo)

Arin-myta-m

Gzplytle+l

&e.

the value of an annuity for the existence of the last

survivor of z and y.

AS
P
Hou a ol

Peg

~ ga.
JF eff ees
||

= ditto 2, y, and z.
a = ditto 2, ¥, 2, w, &e.
a; = the value of an annuity payable so long as any two
of z, y, and z are jointly in existence.
Oiges™ = the value of an annuity payable so long as the last
m survivors of the lives z, y, z, w... are alive.
a = the value of an annuity to continue until z shall have

died and the joint existence of y and z have failed.

244

Gay sn)eer) = the value of an annuity to continue until the joint
existence of z, y,.... shall have failed, and the
joint existence of u, r,..... have also failed.

@,,5 = the value of an annuity to continue so long as # and
either y or z shall be jointly in existence.

@= ~ == the value of an annuity to continue so long as one
of the lives z, y, and one of the lives z, w, shall be
jointly in existence.

Gey. == the value of an annuity to continue so long as z and
y and either < or to shall be in existence.

Gy = TigT
The meanings of —'a;,, "“"a;,, &c., are obvious without explanation.
N.B. If the annuity is payable in advance, it is called an annutfy-

due and is denoted by the symbol @ instead of a, Thus,
a: 1+ az)

: 1+ zy.
It is obvious that any one of the annuities above considered
may be payable in advance, and in every such case & is to be put

instead of a.

N.B. An annuity which is entered upon at the expiration of
n years (or runs from the end of n years) is called an annuity deferred

nm years.
ai@- == an annuity deferred n years and then)so long as
to continue z shall live.

nldey } = ditto during the joint existence of {® ys

nl@ays L,Y, %
nl@s, |= ditto during the existence of the last {* ¥;
nlQzgs survivor of , Ly Y, %
laz3 = ditto during the joint existence of the two last

survivors of 2, y, 2.
a, == the value of a temporary annuity for the next n
years if x shall live so long.

In"sy\—= the values of temporary annuities for the next n
nays years to depend on the statuses denoted by the suffixes.

245

nlttz == the value of a deferred temporary (or an intercepted)
annuity, to he deferred » years and then continue
é years, subject to the existence of the life z.

mir —_ ditto, subject to the joint existence! y

n tlzys of the lives S 7) Ys &
nlt2zy
| —~
"an = the values of intercepted annuities depending on the
vy statuses denoted by the suffixes.
nl(Qzyew....”

N.B. If the above annuities are-payable in advance, it is under-
stood that a temporary or intercepted annuity-due for ¢ years is one in
respect of which ¢ years’ payments are to be made, thus,

| x =I] + t-1Gz,
[:@cy==1 + je schcy. |

In fact, whatever the status on which the annuity depends, denoting
it by X, we have

(@x=1+ + ax,
or simply, a=] +1.
We also have
nl@z = nix
nlBhzy = n=1,Gay or simply »|8==,..\|a.
nites = voir
n thy = nilt@ey or simply nth gW[p0-

N.B. If any of the above annuities is payable by half-yearly
instalments, this is denoted by appending ®; thus

@ @ .) ” r) ) @ @
Gz, &z; alaz; n&z, n@z; 1 nGey nltazs alti

@ ° e
Azy) e e e e ° e e
@ : : : : ° : :
Azyz) ° e r e e ° e

N.B. If the annuity is payable by quarterly instalments, we
put “ instead of ®; and if by instalments m times a year (or by nvly
instalments), ™.

Obs. It will be noticed that

az=tt+ar, a=t+ar, a= 5+az’.

Also that we have
oBZ=_-j]02, iOS a 202"
As particular cases, we have
ija2=a?, 2Ja'=a'.
Obs. The period for which an annuity is deferred need not be an

exact number of years ; but if the first payment is made after a time
r, which may be either integral or fractional, we shall have

+@: == the value of an annuity on 2, of which a payment
is made first at the time r, and thenceforward
at intervals of a year.

jaf = the value of an annuity on 2, payable half-yearly,
of which a payment (=4) is made first at the
time r, and thenceforward at intervals of half a

lg year.

a } == the values of similar annuities payable respectively

by quarterly and m-ly instalments, the first at the
time r.

N.B. We still have

7|S¢ =r. 1\ae;
ie = alaz,
lax =r. dar’.
These annuities may be temporary
aints » alnOa ’ aln m? alnOe °

Obs. Any of the preceding annuities may be complete, s.e. con-
tinued up to the day of the death of the life or lives, or of the failure
of the status, on which it depends. This is denoted by placing a
small circle above the a ora. Thus:—

Gzy nlx; lade» alte, Gzy, &.

Bs, nléees : :

Gs, nds,
a? ) alas»

Obs. If the annuity is supposed to be payable momently, it is
called a continuous annuity, and is denoted by d. Thus,

247

ds; Ary, &e.
nfs, —salday, :
Incr ays :
nlfer —nltdlsy, °

N.B. In this case it makes no difference whether the annuity is
payable in advance or not.

Obs. Since a term of n years may be considered as @ special kind
of life, such that the probability of living over each of the n years
is certainty, and the probability of dying at the end of the n yearr
is also certainty, denoting such a term by nj, we have

= the value of an annuity during the joint existence
of a life x and a term of n years certain, #.¢.,

als) — 5 temporary annuity for m years on the life z,
= |rdz ;
== the value of an annuity during the existence of the
life x, and the term of n years certain, whichever
ls may endure the longer, t.¢.,
== the value of an annuity for n years certain, and 60
much longer as 2 may live.
Similarly

a = the value of an annuity for n years certain, and 80
much longer as either z or y may live.
— = the value of an annuity for n years certain, and so
much longer as z and y shall both live.
== the value of an annuity to continue for n years
certain, and as much longer as z,y,2... . shall
all live.
=—8 ==: the value of an annuity to continue until the
failure of two of the three statuses represented
by the lives z and y and the term of n years
certain; f.¢., an annuity to cease on the death of
both # and y within n years, or at the expiration
of n years if either or y shall have died within that
term, or on the failure of the joint existence of
#2 and y after that term.
Reverstonary Annuities.
Gye == the value of an annuity on the life of x after the
death of y.

248 _

Oey = do. on the joint lives of + and y after the
. death of z.
Qj; = =do. on the life of the last survivor of z and
y after the death of z.
Qnic = do. on the life of # after the failure of the
joint existence of y and z,
Bic = do. on the life of x after the death of both
y and z..
Q'm| = do. to commence on the death of y, and to
° continue payable to the end of » years
from the present time or to the death
of z, whichever shall first happen.*

a= do, do. do. whichever shall
last happen.
Qyal|s = do, on the life of 2, to commence on the

death of y or at the expiration of n
years, whichever may first happen.
a= = do. on the life of z, to commence on the
death of y or at the expiration of n
years, whichever shall last happen. *

Obs. These annuities may be deferred, temporary, or intercepted.

Thus
nldy!z, laQy'x nlt2ylx- .

N.B. In explanation of the principle of this notation, it is to be
observed that the letters or numbers placed at the right-hand lower
corner of the principal symbol for an annuity or assurance, refer
exclusively to lives; while letters or numbers placed at the left-hand
lower corner refer exclusively to terms of years: also, that a letter or
number placed on the left-hand side of the upright stroke, |, denotes
a, life or a term of years, as the case may be, for which the benefit is
deferred.

Obs. Deferred and intercepted annuities may be considered as a
special kind of reversionary annuities, if we consider a term of years
as a special kind of life, as explained above. Thus

nl@xz=Aallz-
Also, a'ttz=Apllntie 5
* This benefit must not be confounded with the annuity to commence on the death of
y, and to continue payable for # years if x shall live so long; for in that case the term of »
yzars does not begin to run until the death of y,

249

that is, an annyity on the life of x, deferred n years and then to
continue ¢ years, is equal to an annuity for so much of the joint
existence of the life z and the term of n+¢ years reckoned from the
present time, as remains after the termination of n years.

N.B. In the above reversionary annuities, it is assumed that the
first payment of the annuity is made at the end of the year in which
the first life (or status) fails; and that the last payment is made at the
end of the year preceding that in which the life (or status) during
which the annuity is to be enjoyed, fails.

If, however, the annuity is to run from the day of the failure of
the first life (or status) to the day of the failure of the second life (or
status) ; as, for example, from the day of the death of y up to the
day of the death of z, tho first payment being made one year after the
death of y; the annuity is by analogy called complete, and is denoted by

Aylz; Galzy: Gys|2 ’ de.

If the tirst payment falls due on the day of the failure of the first
life (or status), or the annuity is payable in advance, being still sup-
posed to be continued to the day of the failure of the second status,
it is denoted by

Gl, Bley, a,,|- ’ ce, ;
If these annuities are payable half-yearly, quarterly, dc., this
will be denoted in the same way as for annuities on single lives.
Thus,

dyz = the value of an annuity on the life of 2, of
which a payment, 4, is first made six months
after the death of y, and thenceforward half-
yearly, a proportionate part of the half-year’s
annuity being payable on the death of .

Avie == the value of an annuity on the life of 2, of
which a payment, 4, is first made on the day
of the death of y, and thenceforward half-
yearly, a proportionate part of the annuity
being payable on the death of z.

Contingent Annuities.

Gilg = the value of an annuity on the life of « after the
death of y, provided he die before z.

Or ae == the value of an annuity on the life of 2 after the
death of y, provided he die after z.

jie = the value of an annuity on « after the death of Up
provided he die within n years. 9

K

OY

ax)
Boo
n|@oo
“al
or aallral

250

Survwvorship Annutties.

the value of an annuity to commence on the death
of the first of z and y, and continue to the death
of the second. :

the value of an annuity to commence on the death
of the first of the three lives x, y, and 2, and
continue to the death of the last.

the value of an annuity to commence on the death
of two of the three lives z, y, and z, and continue
to the death of the last.

the value of an annuity to commence on the death
of the first of the three lives x, y, and z, and
continue so long as the other two are jointly in
existence.

Annutties Certain.

the value of an annuity certain for n years.

& perpetuity,

@ perpetuity deferred n years.

a temporary annuity certain of ¢ years deferred
” years.

Annuities on Successive Lives.

N.B. When a letter or number is placed within brackets, ( ), the
status denoted by it is to be nominated at the expiration of the status
denoted by the letter or number immediately preceding.

Only)
Fal)
Bz(y)()

Fecal)»

Gey)

Az\ql)

the value of an annuity on the successive lives of
zand y; y being nominated at the death of z.

the value of an annuity on the life of z, and for
n years longer.

the value of an annuity on the successive lives of
L,Y, %

the value of an annuity for the joint lives of z and
y, and for m years longer if y shall live so long.

the value of an annuity on the life of y, who is
to be nominated on the death of z.

the value of an annuity for n years certain, com-
mencing at the death of x.

251.

Gzy\s) = the value of an annuity on the life of 2, ‘who is to
be nominated on the failure of the successive lives
of # and y.

Gz\a))y = the value of an annuity to commence on the death
of z, and continue for n years if y shall live so
long.

tay = the value of an annuity to commence on the death
of 2, and continue for n years certain and as
much longer as y may live.

Obs. Any of the preceding annuities may be continuous, in which
case d is used instead of a; or they may be complete, in which case g
is used. The last five annuities, when complete, run, like ordinary
reversionary annuities, from the day of the failure of the status, or
life, for which the annuity is deferred, to the day of the failure of the
status, or life, for which it is enjoyed.

Assurances.

A, = the present value of an assurance of 1, payable at
the end of the year in which x shall die; or,

simply, an assurance on 2.
N.B. An assurance is always understood to be for 1 unless

otherwise stated.

Obs. An assurance may, just like an annuity, depend on the
failure of two or more joint lives, or on the failure of the last survivor
of two or more lives. It may also be deferred, temporary, or inter-

cepted. Thus we have assurances
Immediate. Deferred. Temporary. Intercepted.

A; alAz lnAg altAe
Ay : : _
Am : : :
AZ :
A
A=a :

.2 =m the value of an assurance payable on the death of s
or on the death of the last survivor of # and y
whichever may first happen.

Ajyjs = the value of an assurance payable on the failure of

the joint lives z and y or on the death of z which-

ever shal) last happen.

1A, — Ags Ag = Axst-p1

“IA, = Agi ;

“mA, = Asinom :
Similarly for the other assurances.
Ay =: the values of assurances payable at the end of that
An m-part of a year in which the life or status shall
: fail.
A, = the values of continuous assurances, or assurances
Ag payable at the instant when the life or status
: shall fail.
Endowment Assurances.
lpi) == the value of an assurance on the joint duration of

or Ass]

[a Aisy
or Asya]
lp Abizy
or = Azy,s]

or —s_¢Agal

Aity
or Aya

the life z and the term of n years certain ;

the value of an endowment assurance on 2, payable
on his attaining the age 7-+n, or at the end of the
year in which he shall previously die.

the value of an endowment assurance payable in
n years if both 2 and y are then alive, or at the
end of the year in which either 2 or y shall
previously die.

the value of an endowment assurance payable in
n years if either or y is then alive, or at the
end of the year in which the survivor of them
shall previously die,

the value of an endowment assurance deferred ¢
years, that is, an assurance payable if z shall die

_ after ¢ years but before n years, or if he shall

attain the age z-}-n.

Contingent Assurances.

the value of a contingent assurance on a against y,
#.¢, payable at the end of the year in which z shall
die, if he leave y surviving.

the value of an assurance payable at the end of the
year in which z shall die provided he die after y.

258

Obs. These assurances may be deferred, temporary, .or inter-
cepted :—
Deferred. Temporary. ' Intercepted.
Az, bpA dys nL,
nlA,le, InAs, nleAgl
They may also be modified in other respects as ordinary assur-
ances.
Thus
(=) @) (=
A ”? A ay or Ay|-z;
denote assurances payable at the end of that m-part of a year in which

z shall die first and second respectively of the two lives.
And

Al, A? or Ade,
denote continuous contingent assurances payable at the instant when
z shall die first and second respectively of the two lives.

Al a = the value of an assurance on z provided he die
before y or within ¢ years after him.
Als == the value of an assurance on z provided he die more
than ¢ years after the death of y.
Ax) = the value of an assurance on y who is to be nomi-
or al. nated on the death of z.
Aza) ) = the value of an assurance to be payable n years after
or = Aglal } the death of z.
We have : .
A,+ = the value of 1 payable at the end of n years provided
z be then alive ;
== the value of an endowment on x payable in n
= ,H, years;
Aly = the value of 1 payable on the death of z provided

that take place before the end of n years ;
the value of a temporary assurance for n years on z;
[Ac
Just as Al,+A,j=A,,, we have Aly + Aw =A,,],—an equation
expressing that an endowment assurance is the sum of a temporary
assurance and an endowment,

254

Survivorship Assurances involving Three Lives.

Ajsys
A’.y}== the value of an assurance on z second aan
Abn provided he die third veB.
At: ys the value of an assur-} (either Ist or 2nd
A’s:ys}2=4 ance on & provided » lst ,, 3rd »
Ay that he die » 2nd,, 8rd
Amp =: the value of an assurance on the joint lives of , y,
z, provided 2 die first.
Hence Ayip= Ag: also A®?’, a= Aysle.
Ay first,
Ags = the value of an assurance on the last sur-4 second,
Aa vivor provided that x die third.
|
Hence Abi = m= Apis.
Apt rs
Ags’ | = the value of an assurance payable on the; second,
Az? second death provided that z die third
3
Hence AS: pag.
Aves : first,
Az. ;t== the value of an assurance payable on the} second,
Ar. death of z or y provided either of them die | third.
Azy, = the value of an assurance on the joint lives of z, y,
T z, provided either z or y die first.
Hence Agu Age
1
1.2
A;j:e) = the value of an assurance payable on Ist or 2nd,
ARs the death of x or y provided either Ist ,, 3rd,
Az, of them die Qnd ,, 8rd

‘) <= the value of an assurance on the last | Arst of the
Anis survivor of # and y provided that second

Aas s die third

Asp Aay
Az iu=A5, ;
3 3
Agu ARe
th f
e value o! an assurance on the Ist or 3rd)of the
last survivor of # and y pro- Ond .. 3rd{ th
vided that z die “oe m
2nd and that z
the value of an assurance Srd die first of

on # provided that he die ond or Srd\ the three.

the value of an(1st or 2nd) and provided that in
assurance on “a Srd the latter case z die
provided he die ” "J first of the three.

the value of an assurance on the last before y,

survivor of # and y provided that z die | after y.

the value of an assurance on x provided that either
zor y die first of the three.

the value of an assurance on 2 provided he die
after y, but within # years after his death.

the value of an assurance on # provided he die
after y, but more than ¢ years after his death.

Annual Premiums.

annual premium for an assurance on 2.

do. do. on joint lives of {* and y,
az, y, and z.
do. do. on last survivor of {* and y,
2, y, and z,
do. do. on 2 provided he die before y.
do. for a temporary assurance for n years on z.
do. do, on joint lives of {* and y;
2, y, and z.
do. do. on last survivor of {: and y;
2, y, and z.
do. do. on # provided he die before y.

256

N.B. If desired, the letter A can be dispensed with in the case
of temporary assurances, by regarding a temporary assurance as &
special kind of contingent assurance, as explained above. Thus,

Pi] = annugl premium for A}, ; or for a temporary assur-

ance for n years on 2.

Pa] == annual premium for Aza] , or for an endowment on

z payable in n years.
Similarly, P,,| will denote the annual premium for A,,, that i is
to say, for an endowment assurance on the life of z, payable in n
years, Thus,

Pes

Poy
Pzyen| }== annual premiums for endowment assurances pay-
Pail able in # years, depending on the life or lives
Pal denoted by the suffixes.

If the premium is payable half-yearly or quarterly, we denote the
half-yearly premium by P, and the quarterly premium by P.
2 4

The symbols P®), P, P), will, by analogy with a, d&c., denote

the total yearly premium when it is payable by half-yearly, quarterly,
or m-ly instalments,

So, also, P will denote the total yearly premium when it is
payable by momently instalments.

If the annual premium is payable for m years only, this is
denoted by prefixing a subscript m to the P. Thus,

mE z=) mE sy) ml zn ) &e.
If it should ever be desired to have the annual premium deferred

for a term of years, this would be expressed by the symbol ,,/P.

Thus, for instance, if the annual premium for an intercepted
assurance were payable at the beginning of each year during the
continuance of the benefit, such premium would be denoted by
nltPaiAc-
Obs. If we wish to denote the annual premium for any other
‘benefit, this may be done by prefixing P to the benefit.
Thus,
P,laz = the annual premium for a deferred annuity.
P,ja, == the same, the first payment of the annuity being in
”m YOars. \

257

If the premium and the annuity in the latter case are both to be
payable half-yearly, the half-yearly premium will be denoted by

P, ja?

Values of Policies.
nVz == value of a policy for 1 on the life of a person of
the age z at entry after it has been n years in
force, the premium being just due and unpaid.

nVey = do. on the joint lives of {: and 9,

nV sve 2,Y; and 2.
pled = do. on the last survivor of { and y,

nVon az, y, and 2.

nVeml = value of the endowment assurance Azm| after it has
been in force n years.
Asya;
zy ml
aVzy = Value of a contingent policy on z against y after it
has been n years in force.

If n years and a fractional part, r, of another year have elapsed,
the value of the policy may be denoted by ayrVe-

The value at the end of » years, the premium being just paid,
may be denoted by ni Vz.

= do of the endowment assurance